Figure 1.
Stability region for RCM.
Citation: Seoyoung Yoon, Yong-Ku Kim. The role of immunity and neuroinflammation in genetic predisposition and pathogenesis of Alzheimer's disease[J]. AIMS Genetics, 2015, 2(3): 230-249. doi: 10.3934/genet.2015.3.230
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Fractional calculus (FC) is not a new research area; in reality, it has almost the same history as classical calculus. It motivates the study of derivatives and integrals of fractional order (FO). The history of FC was started in 1695 and appreciated during last few decades, when L'Hopital's question to Leibniz about the differentiation of $ \dfrac{d^{\frac{1}{2}}}{dx}(x) $ [1]. For this, at that time, dated 30 September 1695, Leibniz responded to L'Hopital that "This is an apparent Paradox from which, one day, useful consequences will be drawn" and that was the birth of FC. After that, this concept of fractional differentiation, fractional integration, and its application has found in many numerous areas of the research field of sciences and engineering, especially in control engineering, electromagnetism, signal processing, fluid mechanics, diffusion process, biosciences, statistical, and continuum mechanics and many more. Also, from time to time, FC has been generalized by many researchers and mathematicians, namely Euler, Laplace, Lagrange, Fourier, Riemann, and many others. The differentiation of FO $ \beta > 0 $ has various definitions. But Riemann Liouville (RL) derivative and Caputo's derivative are the old and most commonly used. The CFD is used for our work as it has some advantage in dealing with the IVP of fractional differential equation of non-integer order $ \beta > 0 $. Also, many authors [2,3] have given the existence and uniqueness conditions of the IVP for FDEs.
In the study, it has been found that most of the IVP of FDEs don't have an exact scheme to find the solution, especially for non-linear FDEs. So, it becomes a challenging situation for researchers to establish some methods for finding the analytical solutions of FDEs. Therefore, many researchers have suggested several methods numerically for extended approximate solutions of integer differential equations into fractional differential equations. These schemes incorporates: fractional differential transform scheme, Adomain decomposition scheme [4], variational iteration method [5], spectral collocation scheme [6], fractional finite difference scheme [7], fractional Adams scheme [8], homotopy perturbation scheme [9], homotopy analysis scheme [10], extrapolation method [11], and many others.
In current decades, the IVP of FDEs used as a weapon to solve the various mathematical models, the epidemic model, the disease model, the dynamical system model, and many others. In recent times, Tong et al. [12] proposed fractional EM and fractional IEM, which are the generalization of classical EM and IEM for first-order IVP of FDEs. That EM has a linear convergence rate while IEM has a quadratic convergence rate. In [13] (Chapter 06), C. Milici et al. study several numerical methods for FO systems. In which, they proposed variational iteration, least squares, Euler's, and Runge-Kutta methods for the system of FDE in the CFD sense. In [14], Kumar et al. suggested a numerical scheme to demonstrate the numerical behavior of the IVP of FDEs in which one of the methods is midpoint point whose convergence rate is quadratic. In [15], Muhammad et al. developed a two-stage generalized Rk2 scheme of second order in the CFD sense. These all referred works motivate us to establish more accurate schemes to solve the IVP of FDE in the CFD sense. Also, our objective is to show, based on a few concrete examples and a few application models, that FDEs can model the physical problem more effectively than ODEs. Recently, many research article has been found that solve the real-world phenomenon in FDEs [16,17,18]. This work proposes a fractional RCM scheme for IVP of linear and non-linear FDEs of order $ \beta \in [0, 1] $. This scheme has a cubic convergence rate. It is the generalization of classical RCM, developed by Anthony Ralston [19,20]. This proposed scheme has a more accurate approximation compared to existing fractional EM, IEM, Midpoint method (MPM), and many others for the IVP of FDE:
|
$ CDβx+0u=g(x,u),withinitialconditionu(x0)=u0,andx∈(x0,xEnd]. $
|
(1.1) |
Here, $ _{\mathcal C}\mathcal D^{\beta}_{x_0^+} $ indicates the CFD of arbitrary order $ \beta $ where $ \beta \in [0, 1] $.
By using the proposed work, our main goal is to establish a novel study which is more accurate and appropriate in order to derive the approximate solution of IVP of FDEs (1.1). This scheme incorporates the algorithm, order of convergence, stability, and few numerical examples including the application to WPG model. The scheme also recognize as a type of Runge-Kutta third order method (RK3) and this is explicitly familiar with the term \enquote{One-Step} method.
The presentation of the paper is designed as follows. In Section 2, we provide some preliminary definitions and properties of fractional derivatives and integrals. Next, in Section 3, we suggest our methodology briefly and its order of convergence by following some essential lemma and theorem. Then, in Section 4, we establish the stability of the concerned scheme. After that, in Section 5, we implemented the proposed method on a few examples of linear and nonlinear IVP of FDEs in the CFD frame. In Section 6, we solved World Population Growth (WPG) model via the suggested scheme with the comparison of EM and IEM. Finally, in Section 7, we conclude our methodology with some essential annotations.
This section focuses on some basic definitions of fractional derivatives and integrals, properties, and valuable results in the RL and Caputo derivative sense. This section will be helpful in our this advancement work as this will arise the generalization of ordinary calculus [21,22,23,24,25].
Definition 2.1. [26] The FO integral in the sense of RL derivative for the function $\kappa:[a, b]\rightarrow \mathbb{R}$ of arbitrary order $\beta>0$ are
|
$ Jβa+κ(ζ)=1Γ(β)∫ζaκ(p)(ζ−p)1−βdp,ζ>a,andJβb−κ(ζ)=1Γ(β)∫bζκ(p)(p−ζ)1−βdp,ζ<b, $
|
called the left and right RL fractional integral respectively. Here $\Gamma(\beta)$ denotes the Euler's Gamma function.
Definition 2.2. [26,27] The FO derivatives in the RL sense for the function $\kappa:[a, b]\rightarrow \mathbb{R}$ of order $\beta>0$ are
|
$ RLDβa+κ(ζ)=1Γ(n−β)dndζn∫ζa(ζ−p)n−β−1κ(p)dp,ζ>a,andRLDβb−κ(ζ)=(−1)nΓ(n−β)dndζn∫bζ(p−ζ)n−β−1κ(p)dp,ζ<b, $
|
called the left and right RL fractional derivative respectively, where $n = 1+[{\beta}]$ and $[\beta]$ indicate the integral part of $\beta$. Particularly, if we take $0 < \beta < 1 $, then
|
$ RLDβa+κ(ζ)=1Γ(1−β)ddζ∫ζa(ζ−p)−βκ(p)dp,ζ>a,andRLDβb−κ(ζ)=−1Γ(1−β)ddζ∫bζ(p−ζ)−βκ(p)dp,ζ<b, $
|
are called the left and right RL derivatives of order $\beta$, where $0 < \beta < 1$.
Definition 2.3. [26] The FO derivatives in the Caputo sense for the function $\kappa:[a, b]\rightarrow \mathbb{R}$ of order $\beta>0$ are
|
$ CDβa+κ(ζ)=1Γ(n−β)∫ζa(ζ−p)n−β−1κ(n)(p)dp,ζ>a,andCDβb−κ(ζ)=(−1)nΓ(n−β)∫bζ(p−ζ)n−β−1κ(n)(p)dp,ζ<b, $
|
called the left and right Caputo derivative respectively, where $n = 1+[{\beta}]$.
Particularly, if we take $0 < \beta < 1$, then
|
$ CDβa+κ(ζ)=1Γ(1−β)∫ζa(ζ−p)−βκ′(p)dp,ζ>a,andCDβb−κ(ζ)=−1Γ(1−β)∫bζ(p−ζ)−βκ′(p)dp,ζ<b, $
|
are called the left and right FO Caputo derivatives of order $\beta$, where $0 < \beta < 1$.
The relation between the FO derivative in Caputo fractional derivative and Riemann-Liouville fractional derivative is
|
$ CDβa+κ(ζ)=RLDβa+κ(ζ)−n−1∑k=0κk(a)(ζ−a)k−βΓ(k−β+1),andCDβb−κ(ζ)=RLDβb−κ(ζ)−n−1∑k=0κk(b)(b−ζ)k−βΓ(k−β+1), $
|
where $ n = 1+[{\beta}] $.
Definition 2.4. [26] The one and two parameter Mittag-Leffler function are defined by,
|
$ Eβ(ζ)=∞∑n=0ζnΓ(βn+1),β,ζ∈C;Re(β)>0,andEβ,γ(ζ)=∞∑n=0ζnΓ(βn+γ),β,γ,ζ∈C;Re(β),Re(γ)>0, $
|
respectively. If $\beta \in \mathbb{C}$ with $Re(\beta)>0$, then the series $E_\beta(\zeta)$ is convergent for all $\zeta\in \mathbb{C}$. Similarly, if $\beta, \gamma \in \mathbb{C} $ with $Re(\beta), Re(\gamma)>0 $, then the series $E_{\beta, \gamma}(\zeta)$ is convergent for all $\zeta\in \mathbb{C}$.
Lemma 2.1. [26] If $\beta$, $\gamma \geq 0$, and $\Phi \in L_1\left[a, b\right]$, then
| $ J^\beta_{a^+}J^\gamma_{a^+}\Phi = J^{\beta+\gamma}_{a^+}\Phi,\; \; J^\beta_{b-}J^\gamma_{b-}\Phi = J^{\beta+\gamma}_{b-}\Phi, $ |
holds everywhere on the interval $\left[a, b\right]$. If $\Phi(x)\in C\left[a, b\right]$ or $1\leq\beta+\gamma$, then identity holds everywhere on the interval $\left[a, b\right]$.
Lemma 2.2. [28] If $\Phi\in C^n[a, b]$, $a < b$ and $n\in \mathbb{N}$. Moreover, If $\beta_1, \beta_2 > 0$ be such that, $\exists$ some $k\in \mathbb{N}$ with $k\leq n$ and $\beta_1$, $\beta_1+\beta_2 \in [k-1, k]$. Then,
| $ _{\mathcal C}{\mathcal D}^{\beta_1}_{a^+} {}_{\mathcal C}{\mathcal D}^{\beta_2}_{a^+}\Phi = \; _{\mathcal C}{\mathcal D}^{\beta_1+\beta_2}_{a^+}\Phi. $ |
Theorem 2.1. (Existence of IVP of FDE) [12] Let $g(x, u)$ be a function that hold the condition $g(x_0, u(x_0)) = 0$ and also the $g(x, u)$ is continuous on the domain $R:0\leq x-x_0\leq d$, $\left| u-u_0\right| \leq e$, then the FDEs:
|
$ CDβx+0u=g(x,u),withthecondition,u(a)=u0andx∈(x0,xEnd], $
|
(2.1) |
has at least one solution in the interval $0\leq x-x_0\leq \lambda$ with $\lambda = \min\left\{d, \dfrac{e}{M}\right\}$ and $\max\limits_{(x, u)\in \mathbb{R}}{}_{\mathcal C}\mathcal D^{1-\beta}_{x_0^+}g(x, u) < M$.
Theorem 2.2. (Uniqueness of IVP of FDE) [12] Under the hypotheses of Theorem 2.1, and if $g_x(x, u)$ holds the Lipschitz condition in the variable $u$ with Lipschitz constant $0 < L$,
| $ \left|g_x(x,u_1)-g_x(x,u_2) \right|\leq L\left| u_1-u_2\right|, $ |
then the FDEs (2.1) have an unique solution.
In our work, we are concerned about the approximate solution of the IVP for the linear and non-linear FDEs:
|
$ CDβx+0u=g(x,u),withinitialconditionu(x0)=u0,andx∈(x0,xEnd]. $
|
(3.1) |
Here, we assume the derivative is in the CFD sense of FO $ \beta $ where $ \beta \in [0, 1] $. Now, with the help of Lemma 2.2, we apply suitable analogous transformation so that it becomes the classical differential equation. Then, we get the CFD of order $ (1-\beta) $ and the revised IVP of FDEs:
|
$ u′=CD1−βx+0g(x,u),withinitialconditionu(x0)=u0,andx∈(x0,xEnd]. $
|
(3.2) |
Now, to discover the accurate numerical scheme to find the solution of the IVP of FDEs (3.1) is equivalent to locating the accurate numerical scheme for the IVP of FDEs (3.2). For such a precise solution of (3.2), we designed the RCM in fractional derivative operator, which we rename as fractional RCM, and it is more accurate and faster than all other linear and quadratic convergence rates like EM and IEM [12]. Below, we describe the fractional RCM.
For obtaining the approximate solution of FDE (3.2), we consider $ (x_k, u_k) $ be the set points and we make this points accordingly so that the mesh are equally distribute in the whole interval $ [a, b] $ where we set $ x_0 = a $ and $ x_{End} = b $. This idea will be good by selecting an integer which is non-negative say $ N $ and assuming the mesh points. So, By following RCM, we construct the following algorithm:
|
$ {xk=x0+kh,fork=0,1,2,…,Nh=xk+1−xk,uk+1=uk+h9(2l1+2l2+4l3),l1=CD1−βx+0g(x,uk)|x=xk,l2=CD1−βx+0g(x+h2,uk+h2l1)|x=xk,l3=CD1−βx+0g(x+3h4,uk+3h4l2)|x=xk. $
|
Also with the use of Matlab, the RCM algorithm is proven to be an effective and much more accurate compared to linear and quadratic schemes.
Before showing the convergence of our suggested methodology, first we establish few relevant results which will be essential in the establishment of the convergence of the methods.
Lemma 3.1. [12] If $g_x(x, u)$ be a function that hold the condition of Lipschitz in the variable of $u$, for some Lipschitz constant $L>0$,
| $ \left|g_x(x,u_1)-g_x(x,u_2) \right|\leq L \left| u_1-u_2\right|, $ |
and also fulfil the conditions of Theorem 2.1, then $G(x, u) = {}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+}g(x, u)$ also holds the condition of Lipschitz in the of variable $ u$, for some another Lipschitz constant $M>0$,
| $ \left|G(x,u_1)-G(x,u_2) \right|\leq M\left| u_1-u_2\right|. $ |
Lemma 3.2. If the function $ G(x, y)$ follows the condition of Lipschitz for the variable $u$ and also satisfies the condition of Theorem 2.1, then
|
$ τ(x,u)=29G(x,u)+13G(x+h2,u+h2G(x,u))+49G(x+3h4,u+3h4G(x+3h4,u+3h4G(x+h2,u+h2G(x,u)))), $
|
(3.3) |
also satisfies the condition of Lipschitz in the variable of $u$.
Proof.
|
$ |τ(x,u1)−τ(x,u2)|≤29|G(x,u1)−G(x,u2)|+13|G(x+h2,u1+h2G(x,u1))−G(x+h2,u2+h2G(x,u2))|+49|G(x+3h4,u1+3h4G(x+h2,u1+h2G(x,y)))−G(x+3h4,u2+3h4G(x+h2,u2+h2G(x,y)))|≤M|u1−u2|+hM22|u1−u2|+h2M36|u1−u2|=M(1+hM2!+h2M23!)|u1−u2|=Lτ|u1−u2|, $
|
So, $ \left|\tau(x, u_1)- \tau(x, u_2)\right|\leq L_\tau \left|u_1- u_2 \right| $, where $ L_\tau = M\left(1+\frac{hM}{2!}+\frac{h^2M^2}{3!}\right). $
Theorem 3.1. Consider $g_x(x, u)$ be the function that holds the condition of Lipschitz in the variable of $u$ with Lipschitz constant $L>0$,
| $ \left|g_x(x,u_1)-g_x(x,u_2) \right|\leq L\left|u_1-u_2\right|, $ |
and $u(x)$ be the unique solution of IVP of FDEs (3.2).
Let $u_k$ be the generated solution approximation by Ralston's Cubic method for some non-negative integer $N$. Then for each $k = 0, 1, 2, \dots N$,
| $ u\left(x_k\right) -u_k = O(h^3). $ |
Proof. Let us take RCM iterative formula which is based on $ u_k = u(x_k) $, then we get
|
$ ˉuk+1=u(xk)+h9[2CD1−βx+0g(x,uk)|x=xk+3CD1−βx+0g(x+h2,uk+h2CD1−βx+0g(x,uk)|x=xk)|x=xk4CD1−βx+0g(x+3h4,uk+3h4CD1−βx+0g(x+h2,uk+h2CD1−βx+0g(x,uk)|x=xk)|x=xk)|x=xk]. $
|
Assuming, $G(x, u) = {}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g(x, u)$, then
|
$ ˉuk+1=u(xk)+h9[2G(xk,uk)+3G(xk+h2,uk+h2u′(xk))+4G(xk+3h4,uk+3h4G(xk+h2,uk+h2u′(xk)))]=u(xk)+2h9G(xk,uk)+h3[G(xk,uk)+(h2Gx(xk,uk)+h2Gu(xk,uk)u′(xk))+12!((h2)2Gxx(xk,uk)+h22u′(xk)Gxu(xk,uk)+(h2)2(u′(xk))2Guu(xk,uk))+O(h3)]+4h9[G(xk,uk)+(3h4Gx(xk,uk)+3h4(G(xk,uk)+h2Gx(xk,uk)+h2u′(xk)Gu(xk,uk))Gu(xk,uk))+12!(3h4)2(Gxx(xk,uk)+2Gxu(xk,uk)(G(xk,uk)+O(h))+Guu(xk,uk)(G(xk,uk)+O(h))2)+13!(3h4)3(Gxxx(ξ,η)+3(G(xk,uk)+O(h))Gxxu(ξ,η)+3(G(xk,uk)+O(h))2Gxuu(ξ,η)+(G(xk,uk)+O(h))3Guuu(ξ,η))]=u(xk)+hu′(xk)+h22[Gx(xk,uk)+u′(xk)Gu(xk,uk)]+h33![Gxx(xk,uk)+2u′(xk)Gxu(xk,uk)+Guu(xk,uk)(u′(xk))2+Gx(xk,uk)Gy(xk,uk)+u′(xk)(Gu(xk,uk))2]+O(h4)=u(xk)+hu′(xk)+h22u″(xk)+h33!u‴(xk)+O(h4). $
|
(3.4) |
By using the Taylor's series, the exact form of the solution will be:
|
$ u(xk+1)=u(xk)+hu′(xk)+h22!u″(xk)+h33!u‴(xk)+h44!u⁗(xk)+… $
|
(3.5) |
Now, from the expression (3.4) and (3.5), we obtained $ \left|u(x_{k+1})-\bar u_{k+1}\right| = O(h^4) $. So, we get
| $ \left| u(x_{k+1})-\bar u_{k+1} \right|\leq Kh^4. $ |
Let us assume that,
|
$ τ=29G(x,u)+13G(x+h2,u+h2G(x,u))+49G(x+3h4,u+3h4G(x+3h4,u+3h4G(x+h2,u+h2G(x,u)))), $
|
then using the stated Lemmas 3.1 and 3.2, we have
|
$ |ˉuk+1−uk+1|≤|u(xk)−uk|+h|τ(xk,u(xk))−τ(xk,uk)|≤(1+hLτ)|u(xk)−uk|. $
|
Now,
|
$ |u(xk+1)−uk+1|≤|u(xk+1)−ˉuk+1|+|ˉuk+1−uk+1|≤Kh4+(1+hLτ)|u(xk)−uk|. $
|
So, we get the estimation, $ \left| E_{k+1}\right| = (1+hL_\tau)\left|E_k\right|+Kh^4. $
Thus, we get the recursion relation,
| $ \left|E_k \right|\leq (1+hL_\tau)^k\left|E_0\right|+\frac{Kh^3}{L_\tau}\left[\left(1+hL_\tau\right)^k-1\right]. $ |
As, $ x_k-x_0 = kh \; {\rm{and}}\; E_0 = 0\; {\rm{then}}, \; \; (1+hL_\tau)^k\leq e^{khL_\tau} = \phi_\tau. $
So, we have $ \left|E_k \right|\leq \frac{Kh^3}{L_\tau}\; (\phi_\tau -1). $
Therefore, $ \left|u(x_k)-u_k\right| = O(h^3). $
This conclude that RCM has a cubic convergence rate.
In this section, let us look at the numerical stability of our proposed scheme. Consider the IVP of FDE 3.2 in the simplest form by assuming $ G(x, u) = {}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+}g(x, u) $ as,
|
$ u′=G(x,u),withinitialconditionu(x0)=u0,andx∈[x0,xEnd]. $
|
(4.1) |
The numerical solution of the RCM scheme is given by the formula:
|
$ {uk+1=uk+h9(2l1+2l2+4l3),l1=G(xk,uk),l2=G(xk+h2,uk+h2l1),l3=G(xk+3h4,uk+3h4l2). $
|
In the simplest form, the concept of absolute stability [29,30] is based on the analysis of the behavior, according to the values of the step h, of the numerical solutions of the model equation:
| $ u'(x) = \mu u. $ |
The linearized equation uses $ G(x, u) = \mu u $. In this case,
|
$ l1=μuk,l2=μ(1+μh2)uk,l3=μ(1+3μh4(1+μh2))uk. $
|
These combine to form,
| $ u_{k+1} = \left[1+(h\mu)+\frac{(h\mu)^2}{2}+\frac{(h\mu)^3}{6}\right]u_k = \xi(h\mu)u_k. $ |
Let us put $ h\mu = z $, then the absolute stability region is the set
| $ \{z \in \mathbb{C}: |\xi(h\mu)|\leq 1\}. $ |
Let us examine the stability region of the numerical scheme RCM. Here, the stability region is the set of points such that $ |\xi(h\mu)|\leq 1 $, which is shown in the Figure 1. Note that for stability, the choice of $ h $ must guarantee that $ |h\mu| $ is inside the region. Furthermore, the real parts must be nonnegative for stability (or marginal stability). Therefore the regions to the left of the imaginary axis are the only ones of relevance.
In this section, we illustrate how our proposed scheme operates in practice. We consider few examples of linear as well as non-linear IVP for FDEs in CFD form and solved numerically using the RCM scheme. First we compare the numerical solution with the analytical ones and then compare with the existing EM and IEM schemes. All the numerical computation have been carried out in MATLAB R2016a version. Now, before proceeding of numerical examples, we define few terminology.
The absolute error used in the table is defined as, $ e_N = \max\limits_{j\in\mathbb{Z}}|u(x_j)-u_N(x_j)|. $ The used estimated oder of convergence (EOC) is defined by the quantity as,
| $ EOC = \log_2\frac{\|u-u_N\|_{\infty}}{\|u-u_{2N}\|_{\infty}}, $ |
where $ u_N $ and $ u_{2N} $ are the approximate solution at two distinct grids, with step length $ h $ and $ \frac{h}{2} $, respectively.
In this subsection, we consider four examples of fractional IVPs of FDEs. In these four examples, the Examples 5.1 and 5.2 are linear, Examples 5.3 and 5.4 are non-linear. Here, we consider in Examples 5.1–5.3, the FDE has exact solution and Example 5.4, the FDE has no exact solution. These all FDEs are solved using the proposed fractional RCM including the proper comparison with the existing fractional EM and IEM.
Example 5.1. Consider the following fractional linear initial value problem (IVP) of FDE:
|
$ CDβ0+u=x3,0<x≤1,u(0)=0. $
|
(5.1) |
For $ \beta = \frac{1}{2} $, the exact solution of (5.1) is,
| $ u(x) = \dfrac{\Gamma{(4)}}{\Gamma{(4.5)}}x^{3.5} . $ |
Using our suggested scheme for $ \beta = \frac{1}{2} $ and with step length $ h = \frac{1}{10} $, the numerical solution of (5.1) are graphically represented in the Figure 2 and their exact solution, approximate solution are illustrated in the Table 1. Together with this, the absolute error visualization is indicated in Figure 3. The EOC and CPU performance of the schemes is tabulated in Table 2, and its plot is illustrated in Figure 4.
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 0.10 | 0.000163 | 0.000000 | 0.000163 | 0.000285 | 0.000122 | 0.000157 | 0.000006 |
| 0.20 | 0.001845 | 0.000571 | 0.001275 | 0.002186 | 0.000340 | 0.001837 | 0.000009 |
| 0.30 | 0.007628 | 0.003801 | 0.003828 | 0.008250 | 0.000622 | 0.007617 | 0.000011 |
| 0.40 | 0.020879 | 0.012700 | 0.008179 | 0.021835 | 0.000956 | 0.020866 | 0.000013 |
| 0.50 | 0.045593 | 0.030970 | 0.014624 | 0.046927 | 0.001334 | 0.045578 | 0.000015 |
| 0.60 | 0.086305 | 0.062885 | 0.023420 | 0.088057 | 0.001752 | 0.086288 | 0.000017 |
| 0.70 | 0.148030 | 0.113230 | 0.034800 | 0.150237 | 0.002207 | 0.148012 | 0.000018 |
| 0.80 | 0.236223 | 0.187245 | 0.048978 | 0.238919 | 0.002696 | 0.236203 | 0.000020 |
| 0.90 | 0.356743 | 0.290592 | 0.066151 | 0.359959 | 0.003216 | 0.356722 | 0.000021 |
| 1.00 | 0.515830 | 0.429326 | 0.086505 | 0.519596 | 0.003766 | 0.515808 | 0.000022 |
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| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.08650 | --- | 0.00005 | 0.00377 | --- | 0.00006 | 0.00002 | --- | 0.00008 |
| 20 | 0.04419 | 0.96891 | 0.00011 | 0.00094 | 2.00107 | 0.00024 | 0.00000 | 2.97066 | 0.00032 |
| 40 | 0.02233 | 0.98472 | 0.00029 | 0.00024 | 2.00039 | 0.00067 | 0.00000 | 2.97987 | 0.00100 |
| 80 | 0.01123 | 0.99242 | 0.00006 | 0.00034 | 2.00014 | 0.00068 | 0.00000 | 2.98607 | 0.00115 |
| 160 | 0.00563 | 0.99623 | 0.00117 | 0.00001 | 2.00005 | 0.00194 | 0.00000 | 2.99030 | 0.00284 |
| 320 | 0.00282 | 0.99812 | 0.00244 | 0.00000 | 2.00002 | 0.00469 | 0.00000 | 2.99321 | 0.00578 |
| 640 | 0.00141 | 0.99906 | 0.00490 | 0.00000 | 2.00001 | 0.01049 | 0.00000 | 2.99524 | 0.01400 |
| 1280 | 0.00071 | 0.99953 | 0.00862 | 0.00000 | 2.00000 | 0.01811 | 0.00000 | 2.99664 | 0.02656 |
| 2560 | 0.00035 | 0.99977 | 0.01213 | 0.00000 | 2.00000 | 0.02224 | 0.00000 | 2.99744 | 0.04221 |
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In the Figure 2, it represents the comparison between the exact and numerical result of our proposed scheme RCM with EM and IEM. Table 1 indicates the exact value, approximate value and absolute error of EM, IEM, and our proposed scheme RCM in which we observe that the numerical solution of RCM are more accurate to the exact solution. Figure 3 represents the absolute error of EM, IEM and our proposed scheme RCM, in which we notice that RCM has minimum absolute error in the comparison of EM and IEM, while IEM has second order of convergence. The order of convergence of our suggested scheme RCM, EM, IEM is tabulated in the Table 2 and graphically shown in the Figure 4. From there, it is clear that the order of EM is linear, the order of IEM is quadratic and the order of RCM is cubic. So, RCM is better than the EM and IEM. Next, in the Figure 4, the blue line indicates the linear convergence of EM, the magenta line indicates the quadratic convergence of IEM, and the green line indicates the cubic convergence of RCM respectively.
From our stated Example 5.1, the conclusion is that RCM is much more accurate in the comparison of the EM and IEM. Similar conclusion can also be drawn in the Examples 5.2 and 5.3.
Example 5.2. Consider the following linear IVP of FDE [31]:
|
$ CDβ0+u=u,0.1<x≤1,u(0.1)=Eβ((0.1)β). $
|
(5.2) |
For $ \beta = \frac{1}{2} $, the exact solution of (5.2) is,
| $ u(x) = E_\beta(x^\beta). $ |
Using our suggested scheme for $ \beta = \frac{1}{2} $ and with step length $ h = \frac{1}{10} $, the numerical solution of (5.2) are graphically represented in the Figure 5 and their exact solution, approximate solution are illustrated in the Table 3. Together with this, the absolute error visualization is indicated in Figure 6. The EOC and CPU performance of the schemes is tabulated in Table 4, and its plot is illustrated in Figure 7.
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.10 | 1.486763 | 1.486763 | 0.000000 | 1.486763 | 0.000000 | 1.486763 | 0.000000 |
| 0.20 | 1.799017 | 1.813852 | 0.014835 | 1.803337 | 0.004320 | 1.799343 | 0.000326 |
| 0.30 | 2.107699 | 2.119910 | 0.012211 | 2.113254 | 0.005555 | 2.108071 | 0.000372 |
| 0.40 | 2.430043 | 2.433687 | 0.003644 | 2.436248 | 0.006205 | 2.430428 | 0.000385 |
| 0.50 | 2.774286 | 2.765897 | 0.008389 | 2.780962 | 0.006676 | 2.774675 | 0.000389 |
| 0.60 | 3.146213 | 3.123114 | 0.023099 | 3.153299 | 0.007086 | 3.146603 | 0.000390 |
| 0.70 | 3.550803 | 3.510572 | 0.040230 | 3.558285 | 0.007482 | 3.551193 | 0.000390 |
| 0.80 | 3.992836 | 3.933086 | 0.059750 | 4.000723 | 0.007887 | 3.993226 | 0.000390 |
| 0.90 | 4.477185 | 4.395448 | 0.081737 | 4.485498 | 0.008313 | 4.477573 | 0.000389 |
| 1.00 | 5.008980 | 4.902637 | 0.106343 | 5.017751 | 0.008771 | 5.009367 | 0.000387 |
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| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.09645 | --- | 0.00006 | 0.00715 | --- | 0.00007 | 0.00029 | --- | 0.00010 |
| 20 | 0.04997 | 0.94871 | 0.00012 | 0.00183 | 1.96664 | 0.00016 | 0.00004 | 2.89943 | 0.00022 |
| 40 | 0.02544 | 0.97401 | 0.00028 | 0.00046 | 1.99007 | 0.00031 | 0.00000 | 2.98186 | 0.00054 |
| 80 | 0.01284 | 0.98703 | 0.00033 | 0.00012 | 1.99737 | 0.00059 | 0.00000 | 3.00258 | 0.00090 |
| 160 | 0.00645 | 0.99354 | 0.00064 | 0.00003 | 1.99933 | 0.00181 | 0.00000 | 3.00457 | 0.00370 |
| 320 | 0.00323 | 0.99678 | 0.00229 | 0.00001 | 1.99983 | 0.00451 | 0.00000 | 3.00313 | 0.00711 |
| 640 | 0.00162 | 0.99839 | 0.00238 | 0.00000 | 1.99996 | 0.00505 | 0.00000 | 3.00178 | 0.01928 |
| 1280 | 0.00081 | 0.99920 | 0.00589 | 0.00000 | 1.99999 | 0.00680 | 0.00000 | 3.00095 | 0.02912 |
| 2560 | 0.00040 | 0.99960 | 0.00682 | 0.00000 | 2.00000 | 0.01662 | 0.00000 | 3.00009 | 0.04033 |
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Example 5.3. Consider the following non-linear IVP of FDE:
|
$ CDβ1+u=(35√π32)u67,1<x≤2,u(1)=1. $
|
(5.3) |
For $ \beta = \frac{1}{2} $, the exact solution of (5.3) is,
| $ u(x) = x^{3.5}. $ |
Using our suggested scheme for $ \beta = \frac{1}{2} $ and with step length $ h = \frac{1}{10} $, the numerical solution of (5.3) are graphically represented in the Figure 8 and their exact solution, approximate solution are illustrated in the Table 5. Together with this, the absolute error visualization is indicated in Figure 9. The EOC and CPU performance of the schemes is tabulated in Table 6, and its plot is illustrated in Figure 10.
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 1.00 | 1.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 1.10 | 1.395965 | 1.350000 | 0.045965 | 1.391837 | 0.004127 | 1.395743 | 0.000221 |
| 1.20 | 1.892929 | 1.783674 | 0.109255 | 1.883452 | 0.009478 | 1.892442 | 0.000488 |
| 1.30 | 2.504965 | 2.312825 | 0.192141 | 2.488830 | 0.016135 | 2.504165 | 0.000800 |
| 1.40 | 3.246745 | 2.949869 | 0.296875 | 3.222565 | 0.024180 | 3.245583 | 0.001161 |
| 1.50 | 4.133514 | 3.707821 | 0.425693 | 4.099827 | 0.033687 | 4.131942 | 0.001571 |
| 1.60 | 5.181076 | 4.600266 | 0.580810 | 5.136346 | 0.044730 | 5.179043 | 0.002033 |
| 1.70 | 6.405768 | 5.641347 | 0.764422 | 6.348392 | 0.057377 | 6.403222 | 0.002546 |
| 1.80 | 7.824449 | 6.845745 | 0.978704 | 7.752755 | 0.071694 | 7.821336 | 0.003113 |
| 1.90 | 9.454479 | 8.228665 | 1.225814 | 9.366733 | 0.087746 | 9.450744 | 0.003735 |
| 2.00 | 11.313708 | 9.805821 | 1.507887 | 11.208114 | 0.105594 | 11.309296 | 0.004413 |
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| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 1.50789 | --- | 0.00006 | 0.10559 | --- | 0.00007 | 0.00441 | --- | 0.00011 |
| 20 | 0.80303 | 0.90901 | 0.00027 | 0.02854 | 1.88746 | 0.00045 | 0.00059 | 2.89185 | 0.00064 |
| 40 | 0.41481 | 0.95299 | 0.00046 | 0.00743 | 1.94236 | 0.00091 | 0.00008 | 2.94618 | 0.00117 |
| 80 | 0.21087 | 0.97610 | 0.00074 | 0.00189 | 1.97081 | 0.00093 | 0.00001 | 2.97319 | 0.00251 |
| 160 | 0.10632 | 0.98795 | 0.00109 | 0.00048 | 1.98531 | 0.00465 | 0.00000 | 2.98662 | 0.00889 |
| 320 | 0.05338 | 0.99395 | 0.00326 | 0.00012 | 1.99263 | 0.00564 | 0.00000 | 2.99332 | 0.00976 |
| 640 | 0.02675 | 0.99697 | 0.00729 | 0.00003 | 1.99631 | 0.01485 | 0.00000 | 2.99666 | 0.02247 |
| 1280 | 0.01339 | 0.99848 | 0.01173 | 0.00001 | 1.99815 | 0.03425 | 0.00000 | 2.99833 | 0.03725 |
| 2560 | 0.00670 | 0.99924 | 0.02115 | 0.00000 | 1.99908 | 0.03590 | 0.00000 | 2.99907 | 0.04107 |
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Example 5.4. Consider the following IVP of FDE:
|
$ CDβ0+u=e2x,0<x≤1,u(0)=1. $
|
(5.4) |
With the help of our suggested scheme for $ \beta = \frac{1}{10} $ and with step length $ h = \frac{1}{10} $ and $ h = \frac{1}{20} $, the numerical solution of (5.4) is graphically shown in the Figure 11 and their approximate solutions for $ h = \frac{1}{20} $ is illustrated in the Table 7. In this example, the absolute error is calculated as the difference between the approximate solution at $ N $ grid point and $ 2N $ grid point. The absolute error visualization is indicated in Figure 12 for $ h = \frac{1}{20} $. The EOC and CPU performance of the schemes is tabulated in Table 8, and its plot is illustrated in Figure 13 for $ h = \frac{1}{20} $.
| EM | IEM | RCM | ||||
| $ x $ | $ u_{EM} $ | $ \left|u_{EM}(h)-u_{EM}(\frac{h}{2})\right| $ | $ u_{IEM} $ | $ \left|u_{IEM}(h)-u_{IEM}(\frac{h}{2})\right| $ | $ u_{{RCM}} $ | $ \left|u_{RCM}(h)-u_{RCM}(\frac{h}{2})\right| $ |
| 0.00 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 0.10 | 1.196419 | 0.005035 | 1.206748 | 0.000129 | 1.206575 | 0.000001 |
| 0.20 | 1.436327 | 0.011185 | 1.459271 | 0.000287 | 1.458887 | 0.000001 |
| 0.30 | 1.729350 | 0.018697 | 1.767703 | 0.000479 | 1.767061 | 0.000002 |
| 0.40 | 2.087249 | 0.027872 | 2.144423 | 0.000715 | 2.143466 | 0.000003 |
| 0.50 | 2.524389 | 0.039079 | 2.604549 | 0.001002 | 2.603208 | 0.000005 |
| 0.60 | 3.058312 | 0.052766 | 3.166549 | 0.001353 | 3.164738 | 0.000007 |
| 0.70 | 3.710447 | 0.069484 | 3.852977 | 0.001781 | 3.850593 | 0.000009 |
| 0.80 | 4.506967 | 0.089903 | 4.691383 | 0.002305 | 4.688297 | 0.000011 |
| 0.90 | 5.479839 | 0.114843 | 5.715413 | 0.002944 | 5.711471 | 0.000014 |
| 1.00 | 6.668107 | 0.145305 | 6.966167 | 0.003725 | 6.961180 | 0.000018 |
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| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.28317 | --- | 0.00003 | 0.01489 | --- | 0.00004 | 0.00014 | --- | 0.00005 |
| 20 | 0.14531 | 0.96258 | 0.00007 | 0.00372 | 1.99910 | 0.00007 | 0.00002 | 2.99292 | 0.00010 |
| 40 | 0.07358 | 0.98163 | 0.00011 | 0.00093 | 1.99977 | 0.00012 | 0.00000 | 2.99678 | 0.00019 |
| 80 | 0.03702 | 0.99090 | 0.00017 | 0.00023 | 1.99994 | 0.00024 | 0.00000 | 2.99847 | 0.00065 |
| 160 | 0.01857 | 0.99547 | 0.00028 | 0.00006 | 1.99999 | 0.00049 | 0.00000 | 2.99926 | 0.00075 |
| 320 | 0.00930 | 0.99774 | 0.00034 | 0.00001 | 2.00000 | 0.00063 | 0.00000 | 2.99963 | 0.00096 |
| 640 | 0.00465 | 0.99887 | 0.00104 | 0.00000 | 2.00000 | 0.00164 | 0.00000 | 2.99984 | 0.00190 |
| 1280 | 0.00233 | 0.99944 | 0.00129 | 0.00000 | 2.00000 | 0.00292 | 0.00000 | 2.99947 | 0.00471 |
| 2560 | 0.00116 | 0.99972 | 0.00334 | 0.00000 | 2.00000 | 0.00724 | 0.00000 | 3.00440 | 0.01124 |
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In this section, we consider one application of the real-world phenomenon WPG model in the form of CFD. We solve the model numerically using our proposed scheme. Also, we discuss the benefits of FC using the WPG model.
Example 6.1. Consider the following linear IVP of FDE of WPG model [32],
|
$ CDβt+0N(t)=PN(t),t>t0, $
|
(6.1) |
|
$ N(t0)=N0. $
|
(6.2) |
Here $ \mathcal{N}(t) $ suggest the number of individuals population at any time $ t $ and here our $ P $ represents the production rate where $ P = B-M $, $ B $ indicates the rate of birth, and $ M $ indicates the rate of mortality. Now, if we assume the value of $ \beta = 1 $ then in this case, our corresponding model will be linear population world growth model which is also famous as a classical population world growth model. The exact solution of (6.1) for $ \beta = 1 $ is,
| $ \mathcal{N}(t) = \mathcal{N}_0 \; e^{Pt},\; t > 0. $ |
Where $ \mathcal{N}_0 $ indicates the initial population at the initial time $ t = t_0 $. Our fractional model is better than the classical model from the numerical perspective. We have taken the population database from the year 1920 to 2018, that is around one century from the world population sites https://www.census.gov/data/tables/time-series/demo/international-programs/historical-est-worldpop.html or https://datacommons.org/place/Earth (provided by world bank), and also one is taken by United Nations [33]. These statistical population data match our fractional model scheme for $ \beta = 1.393298754843208 $. Also, The exact solution of (6.1) for the fractional model of population world growth model will be,
| $ \mathcal{N}(t) = \mathcal{N}_0\; E_\beta(Pt),\; t > 0. $ |
Now, in our classical population model scheme, the estimated value of our production rate is $ P \approx 0.013501 $ and for the fractional model scheme, the production rate $ P \approx 0.0034399 $ [34]. So, it has been found that the statistical population date value fit with the our fractional model for $ \beta = 1.3932987548432 $.
The world population data from the year 1920 to 2018 is graphically represented in the Figure 14 and the numerical value of (6.1) for $ \beta = 1 $ and $ \beta = 1.393298754843 $ with the step size $ h = 1 $ year is graphically shown in the Figure 15, tabulated in the Table 9. In addition to this, the absolute error visualization represented in Figure 16. From all the tables and figures, we conclude that our suggested scheme RCM is much more accurate and faster than EM and IEM.
| EM | IEM | RCM | ||||||
| Year(t) | $ \mathcal{N}_{clasical} $ | $ \mathcal{N}_{frac} $ | $ \mathcal{N}_{EM} $ | $ Error $ | $ \mathcal{N}_{IEM} $ | $ Error $ | $ \mathcal{N}_{{RCM}} $ | $ Error $ |
| 1920 | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ |
| 1930 | 2.1289$ \times 10^3 $ | 1.9909$ \times 10^3 $ | 1.9799$ \times 10^3 $ | 1.1060$ \times 10^1 $ | 1.9892$ \times 10^3 $ | 1.7315$ \times 10^0 $ | 1.9904$ \times 10^3 $ | 4.8870$ \times 10^{-1} $ |
| 1940 | 2.4366$ \times 10^3 $ | 2.2169$ \times 10^3 $ | 2.2020$ \times 10^3 $ | 1.4921$ \times 10^1 $ | 2.2152$ \times 10^3 $ | 1.7413$ \times 10^0 $ | 2.2164$ \times 10^3 $ | 4.8880$ \times 10^{-1} $ |
| 1950 | 2.7888$ \times 10^3 $ | 2.5173$ \times 10^3 $ | 2.4987$ \times 10^3 $ | 1.8625$ \times 10^1 $ | 2.5156$ \times 10^3 $ | 1.7390$ \times 10^0 $ | 2.5168$ \times 10^3 $ | 4.8882$ \times 10^{-1} $ |
| 1960 | 3.1919$ \times 10^3 $ | 2.8943$ \times 10^3 $ | 2.8717$ \times 10^3 $ | 2.2615$ \times 10^1 $ | 2.8926$ \times 10^3 $ | 1.7320$ \times 10^0 $ | 2.8938$ \times 10^3 $ | 4.8884$ \times 10^{-1} $ |
| 1970 | 3.6533$ \times 10^3 $ | 3.3561$ \times 10^3 $ | 3.3290$ \times 10^3 $ | 2.7117$ \times 10^1 $ | 3.3544$ \times 10^3 $ | 1.7219$ \times 10^0 $ | 3.3556$ \times 10^3 $ | 4.8885$ \times 10^{-1} $ |
| 1980 | 4.1814$ \times 10^3 $ | 3.9147$ \times 10^3 $ | 3.8824$ \times 10^3 $ | 3.2308$ \times 10^1 $ | 3.9130$ \times 10^3 $ | 1.7090$ \times 10^0 $ | 3.9142$ \times 10^3 $ | 4.8886$ \times 10^{-1} $ |
| 1990 | 4.7858$ \times 10^3 $ | 4.5858$ \times 10^3 $ | 4.5474$ \times 10^3 $ | 3.8363$ \times 10^1 $ | 4.5841$ \times 10^3 $ | 1.6930$ \times 10^0 $ | 4.5853$ \times 10^3 $ | 4.8888$ \times 10^{-1} $ |
| 2000 | 5.4775$ \times 10^3 $ | 5.3885$ \times 10^3 $ | 5.3431$ \times 10^3 $ | 4.5471$ \times 10^1 $ | 5.3869$ \times 10^3 $ | 1.6737$ \times 10^0 $ | 5.3881$ \times 10^3 $ | 4.8889$ \times 10^{-1} $ |
| 2010 | 6.2693$ \times 10^3 $ | 6.3462$ \times 10^3 $ | 6.2923$ \times 10^3 $ | 5.3847$ \times 10^1 $ | 6.3445$ \times 10^3 $ | 1.6505$ \times 10^0 $ | 6.3457$ \times 10^3 $ | 4.8891$ \times 10^{-1} $ |
| 2020 | 7.1755$ \times 10^3 $ | 7.4865$ \times 10^3 $ | 7.4228$ \times 10^3 $ | 6.3739$ \times 10^1 $ | 7.4849$ \times 10^3 $ | 1.6229$ \times 10^0 $ | 7.4860$ \times 10^3 $ | 4.8893$ \times 10^{-1} $ |
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From the year 1920 to June 2018, as per \enquote{The Census Bureau's International Data Base} indicates that the world population data reached around 7.5 billion. This data is very accurate and near to our fractional model for the FO $ \beta = 1.3932987548432 $. So, by our scheme, the numerical population data gives a more precise and appropriate solution than the fractional EM and fractional IEM.
In this paper, the fractional RCM scheme is established for the IVP of FDE in CFD sense for the first time. Here, we do some analogous conversion of CFD of order $ \beta $ into an ODE of integer order one, and then we operate our proposed scheme in the revised problem. The numerical scheme was used directly without consuming the perturbation, linearization, or other assumptions. The convergence analysis and stability analysis of the scheme has been proved. Also, in this work, we demonstrated a comparative numerical study of our proposed scheme with the comparison of the existing scheme fractional EM and fractional IEM for various examples of linear and non-linear FDEs. The scheme also solves one real-world phenomenon: The fractional WPG model. Now, here we conclude the significant benefits of our scheme:
● The fractional RCM scheme has a cubic convergence rate which is slightly faster than the other linear and quadratic convergence methods for the IVP of FDEs.
● The idea of computation of the proposed scheme is better, and we can get the desired approximation with the increment of mesh points.
● With the help of the WPG model, we discovered that FDEs fit the model better than ODE.
In the future, this work may be helpful to solve the IVP of FDE more accurately and effectively.
The first and third author extended their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University, Saudi Arabia.
All authors declare no conflicts of interest.
| [1] | World Health Organization, The global burden of disease: 2004 update. Geneva 27, Switzerland, World Health Organization, 2008. Available from: http: //www.who.int/healthinfo/global_burden_disease/2004_report_update/en/ |
| [2] |
Ferri CP, Prince M, Brayne C, et al. (2005) Global prevalence of dementia: a Delphi consensus study. Lancet 366: 2112-2117. doi: 10.1016/S0140-6736(05)67889-0
|
| [3] |
Prince M, Bryce R, Albanese E, et al. (2013) The global prevalence of dementia: a systematic review and metaanalysis. Alzheimers Dement 9: 63-75.e62. doi: 10.1016/j.jalz.2012.11.007
|
| [4] |
Brookmeyer R, Johnson E, Ziegler-Graham K, et al. (2007) Forecasting the global burden of Alzheimer's disease. Alzheimers Dement 3: 186-191. doi: 10.1016/j.jalz.2007.04.381
|
| [5] |
Hardy JA, Higgins GA (1992) Alzheimer's disease: the amyloid cascade hypothesis. Science 256: 184-185. doi: 10.1126/science.1566067
|
| [6] |
Hardy J, Selkoe DJ (2002) The amyloid hypothesis of Alzheimer's disease: progress and problems on the road to therapeutics. Science 297: 353-356. doi: 10.1126/science.1072994
|
| [7] |
Povova J, Ambroz P, Bar M, et al. (2012) Epidemiological of and risk factors for Alzheimer's disease: a review. Biomed Pap Med Fac Univ Palacky Olomouc Czech Repub 156: 108-114. doi: 10.5507/bp.2012.055
|
| [8] |
Lautenschlager NT, Cupples LA, Rao VS, et al. (1996) Risk of dementia among relatives of Alzheimer's disease patients in the MIRAGE study: What is in store for the oldest old? Neurology 46: 641-650. doi: 10.1212/WNL.46.3.641
|
| [9] |
Raiha I, Kaprio J, Koskenvuo M, et al. (1996) Alzheimer's disease in Finnish twins. Lancet 347: 573-578. doi: 10.1016/S0140-6736(96)91272-6
|
| [10] |
Gatz M, Reynolds CA, Fratiglioni L, et al. (2006) Role of genes and environments for explaining Alzheimer disease. Arch Gen Psychiatry 63: 168-174. doi: 10.1001/archpsyc.63.2.168
|
| [11] | Bergem AL, Engedal K, Kringlen E (1997) The role of heredity in late-onset Alzheimer disease and vascular dementia. A twin study. Arch Gen Psychiatry 54: 264-270. |
| [12] | Gatz M, Pedersen NL, Berg S, et al. (1997) Heritability for Alzheimer's disease: the study of dementia in Swedish twins. J Gerontol A Biol Sci Med Sci 52: M117-125. |
| [13] |
Coon KD, Myers AJ, Craig DW, et al. (2007) A high-density whole-genome association study reveals that APOE is the major susceptibility gene for sporadic late-onset Alzheimer's disease. J Clin Psychiatry 68: 613-618. doi: 10.4088/JCP.v68n0419
|
| [14] | Couzin J (2008) Genetics. Once shunned, test for Alzheimer's risk headed to market. Science 319: 1022-1023. |
| [15] | Holtzman DM, Herz J, Bu G (2012) Apolipoprotein E and apolipoprotein E receptors: normal biology and roles in Alzheimer disease. Cold Spring Harb Perspect Med 2: a006312. |
| [16] | Farrer LA, Cupples LA, Haines JL, et al. (1997) Effects of age, sex, and ethnicity on the association between apolipoprotein E genotype and Alzheimer disease. A meta-analysis. APOE and Alzheimer Disease Meta Analysis Consortium. JAMA 278: 1349-1356. |
| [17] |
Kim KW, Jhoo JH, Lee KU, et al. (1999) Association between apolipoprotein E polymorphism and Alzheimer's disease in Koreans. Neurosci Lett 277: 145-148. doi: 10.1016/S0304-3940(99)00867-8
|
| [18] |
Lambert JC, Amouyel P (2011) Genetics of Alzheimer's disease: new evidences for an old hypothesis? Curr Opin Genet Dev 21: 295-301. doi: 10.1016/j.gde.2011.02.002
|
| [19] |
Bertram L, McQueen MB, Mullin K, et al. (2007) Systematic meta-analyses of Alzheimer disease genetic association studies: the AlzGene database. Nat Genet 39: 17-23. doi: 10.1038/ng1934
|
| [20] |
Rogaeva E, Meng Y, Lee JH, et al. (2007) The neuronal sortilin-related receptor SORL1 is genetically associated with Alzheimer disease. Nat Genet 39: 168-177. doi: 10.1038/ng1943
|
| [21] |
Reitz C, Cheng R, Rogaeva E, et al. (2011) Meta-analysis of the association between variants in SORL1 and Alzheimer disease. Arch Neurol 68: 99-106. doi: 10.1001/archneurol.2010.346
|
| [22] |
Wellcome Trust Case Control Consortium (2007) Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls. Nature 447: 661-678. doi: 10.1038/nature05911
|
| [23] |
Chouraki V, Seshadri S (2014) Genetics of Alzheimer's disease. Adv Genet 87: 245-294. doi: 10.1016/B978-0-12-800149-3.00005-6
|
| [24] |
Jonsson T, Stefansson H, Steinberg S, et al. (2013) Variant of TREM2 associated with the risk of Alzheimer's disease. N Engl J Med 368: 107-116. doi: 10.1056/NEJMoa1211103
|
| [25] | Cruchaga C, Karch CM, Jin SC, et al. (2014) Rare coding variants in the phospholipase D3 gene confer risk for Alzheimer's disease. Nature 505: 550-554. |
| [26] |
Graeber MB, Streit WJ (2010) Microglia: biology and pathology. Acta Neuropathol 119: 89-105. doi: 10.1007/s00401-009-0622-0
|
| [27] |
Ransohoff RM, Perry VH (2009) Microglial physiology: unique stimuli, specialized responses. Annu Rev Immunol 27: 119-145. doi: 10.1146/annurev.immunol.021908.132528
|
| [28] |
Streit WJ (2002) Microglia as neuroprotective, immunocompetent cells of the CNS. Glia 40: 133-139. doi: 10.1002/glia.10154
|
| [29] | Dickson DW, Farlo J, Davies P, et al. (1988) Alzheimer's disease. A double-labeling immunohistochemical study of senile plaques. Am J Pathol 132: 86-101. |
| [30] |
Guillot-Sestier MV, Town T (2013) Innate immunity in Alzheimer's disease: a complex affair. CNS Neurol Disord Drug Targets 12: 593-607. doi: 10.2174/1871527311312050008
|
| [31] |
Coraci IS, Husemann J, Berman JW, et al. (2002) CD36, a class B scavenger receptor, is expressed on microglia in Alzheimer's disease brains and can mediate production of reactive oxygen species in response to beta-amyloid fibrils. Am J Pathol 160: 101-112. doi: 10.1016/S0002-9440(10)64354-4
|
| [32] | Bamberger ME, Harris ME, McDonald DR, et al. (2003) A cell surface receptor complex for fibrillar beta-amyloid mediates microglial activation. J Neurosci 23: 2665-2674. |
| [33] | Frenkel D, Wilkinson K, Zhao L, et al. (2013) Scara1 deficiency impairs clearance of soluble amyloid-beta by mononuclear phagocytes and accelerates Alzheimer's-like disease progression. Nat Commun 4: 2030. |
| [34] |
Walter S, Letiembre M, Liu Y, et al. (2007) Role of the toll-like receptor 4 in neuroinflammation in Alzheimer's disease. Cell Physiol Biochem 20: 947-956. doi: 10.1159/000110455
|
| [35] | Scholtzova H, Chianchiano P, Pan J, et al. (2014) Amyloid beta and Tau Alzheimer's disease related pathology is reduced by Toll-like receptor 9 stimulation. Acta Neuropathol Commun 2: 101. |
| [36] |
Wang LZ, Tian Y, Yu JT, et al. (2011) Association between late-onset Alzheimer's disease and microsatellite polymorphisms in intron II of the human toll-like receptor 2 gene. Neurosci Lett 489: 164-167. doi: 10.1016/j.neulet.2010.12.008
|
| [37] |
Liu Y, Walter S, Stagi M, et al. (2005) LPS receptor (CD14): a receptor for phagocytosis of Alzheimer's amyloid peptide. Brain 128: 1778-1789. doi: 10.1093/brain/awh531
|
| [38] |
Prokop S, Miller KR, Heppner FL (2013) Microglia actions in Alzheimer's disease. Acta Neuropathol 126: 461-477. doi: 10.1007/s00401-013-1182-x
|
| [39] |
Lee CY, Landreth GE (2010) The role of microglia in amyloid clearance from the AD brain. J Neural Transm 117: 949-960. doi: 10.1007/s00702-010-0433-4
|
| [40] |
Stewart CR, Stuart LM, Wilkinson K, et al. (2010) CD36 ligands promote sterile inflammation through assembly of a Toll-like receptor 4 and 6 heterodimer. Nat Immunol 11: 155-161. doi: 10.1038/ni.1836
|
| [41] |
Town T, Nikolic V, Tan J (2005) The microglial ""activation"" continuum: from innate to adaptive responses. J Neuroinflammation 2: 24. doi: 10.1186/1742-2094-2-24
|
| [42] |
Varnum MM, Ikezu T (2012) The classification of microglial activation phenotypes on neurodegeneration and regeneration in Alzheimer's disease brain. Arch Immunol Ther Exp (Warsz) 60: 251-266. doi: 10.1007/s00005-012-0181-2
|
| [43] |
Mantovani A, Sozzani S, Locati M, et al. (2002) Macrophage polarization: tumor-associated macrophages as a paradigm for polarized M2 mononuclear phagocytes. Trends Immunol 23: 549-555. doi: 10.1016/S1471-4906(02)02302-5
|
| [44] |
Goerdt S, Orfanos CE (1999) Other functions, other genes: alternative activation of antigen-presenting cells. Immunity 10: 137-142. doi: 10.1016/S1074-7613(00)80014-X
|
| [45] |
Jimenez S, Baglietto-Vargas D, Caballero C, et al. (2008) Inflammatory response in the hippocampus of PS1M146L/APP751SL mouse model of Alzheimer's disease: age-dependent switch in the microglial phenotype from alternative to classic. J Neurosci 28: 11650-11661. doi: 10.1523/JNEUROSCI.3024-08.2008
|
| [46] |
Hoozemans JJ, Veerhuis R, Rozemuller JM, et al. (2006) Neuroinflammation and regeneration in the early stages of Alzheimer's disease pathology. Int J Dev Neurosci 24: 157-165. doi: 10.1016/j.ijdevneu.2005.11.001
|
| [47] | Martinez FO, Gordon S (2014) The M1 and M2 paradigm of macrophage activation: time for reassessment. F1000Prime Rep 6: 13. |
| [48] |
Simard AR, Soulet D, Gowing G, et al. (2006) Bone marrow-derived microglia play a critical role in restricting senile plaque formation in Alzheimer's disease. Neuron 49: 489-502. doi: 10.1016/j.neuron.2006.01.022
|
| [49] |
Mandrekar S, Jiang Q, Lee CY, et al. (2009) Microglia mediate the clearance of soluble Abeta through fluid phase macropinocytosis. J Neurosci 29: 4252-4262. doi: 10.1523/JNEUROSCI.5572-08.2009
|
| [50] |
Yuyama K, Sun H, Mitsutake S, et al. (2012) Sphingolipid-modulated exosome secretion promotes clearance of amyloid-beta by microglia. J Biol Chem 287: 10977-10989. doi: 10.1074/jbc.M111.324616
|
| [51] |
Maier M, Peng Y, Jiang L, et al. (2008) Complement C3 deficiency leads to accelerated amyloid beta plaque deposition and neurodegeneration and modulation of the microglia/macrophage phenotype in amyloid precursor protein transgenic mice. J Neurosci 28: 6333-6341. doi: 10.1523/JNEUROSCI.0829-08.2008
|
| [52] |
Webster S, Lue LF, Brachova L, et al. (1997) Molecular and cellular characterization of the membrane attack complex, C5b-9, in Alzheimer's disease. Neurobiol Aging 18: 415-421. doi: 10.1016/S0197-4580(97)00042-0
|
| [53] |
Flanary BE, Streit WJ (2004) Progressive telomere shortening occurs in cultured rat microglia, but not astrocytes. Glia 45: 75-88. doi: 10.1002/glia.10301
|
| [54] | Griffin WS, Sheng JG, Royston MC, et al. (1998) Glial-neuronal interactions in Alzheimer's disease: the potential role of a 'cytokine cycle' in disease progression. Brain Pathol 8: 65-72. |
| [55] |
Griffin WS, Stanley LC, Ling C, et al. (1989) Brain interleukin 1 and S-100 immunoreactivity are elevated in Down syndrome and Alzheimer disease. Proc Natl Acad Sci U S A 86: 7611-7615. doi: 10.1073/pnas.86.19.7611
|
| [56] |
Barger SW, Harmon AD (1997) Microglial activation by Alzheimer amyloid precursor protein and modulation by apolipoprotein E. Nature 388: 878-881. doi: 10.1038/42257
|
| [57] | Li Y, Liu L, Barger SW, et al. (2003) Interleukin-1 mediates pathological effects of microglia on tau phosphorylation and on synaptophysin synthesis in cortical neurons through a p38-MAPK pathway. J Neurosci 23: 1605-1611. |
| [58] | Li Y, Liu L, Kang J, et al. (2000) Neuronal-glial interactions mediated by interleukin-1 enhance neuronal acetylcholinesterase activity and mRNA expression. J Neurosci 20: 149-155. |
| [59] |
Sheng JG, Ito K, Skinner RD, et al. (1996) In vivo and in vitro evidence supporting a role for the inflammatory cytokine interleukin-1 as a driving force in Alzheimer pathogenesis. Neurobiol Aging 17: 761-766. doi: 10.1016/0197-4580(96)00104-2
|
| [60] |
Yamanaka M, Ishikawa T, Griep A, et al. (2012) PPARgamma/RXRalpha-induced and CD36-mediated microglial amyloid-beta phagocytosis results in cognitive improvement in amyloid precursor protein/presenilin 1 mice. J Neurosci 32: 17321-17331. doi: 10.1523/JNEUROSCI.1569-12.2012
|
| [61] |
Tarkowski E, Andreasen N, Tarkowski A, et al. (2003) Intrathecal inflammation precedes development of Alzheimer's disease. J Neurol Neurosurg Psychiatry 74: 1200-1205. doi: 10.1136/jnnp.74.9.1200
|
| [62] |
Sheng JG, Jones RA, Zhou XQ, et al. (2001) Interleukin-1 promotion of MAPK-p38 overexpression in experimental animals and in Alzheimer's disease: potential significance for tau protein phosphorylation. Neurochem Int 39: 341-348. doi: 10.1016/S0197-0186(01)00041-9
|
| [63] |
Munoz L, Ralay Ranaivo H, Roy SM, et al. (2007) A novel p38 alpha MAPK inhibitor suppresses brain proinflammatory cytokine up-regulation and attenuates synaptic dysfunction and behavioral deficits in an Alzheimer's disease mouse model. J Neuroinflammation 4: 21. doi: 10.1186/1742-2094-4-21
|
| [64] |
Yoshiyama Y, Higuchi M, Zhang B, et al. (2007) Synapse loss and microglial activation precede tangles in a P301S tauopathy mouse model. Neuron 53: 337-351. doi: 10.1016/j.neuron.2007.01.010
|
| [65] |
Butterfield DA, Swomley AM, Sultana R (2013) Amyloid beta-peptide (1-42)-induced oxidative stress in Alzheimer disease: importance in disease pathogenesis and progression. Antioxid Redox Signal 19: 823-835. doi: 10.1089/ars.2012.5027
|
| [66] |
Thiabaud G, Pizzocaro S, Garcia-Serres R, et al. (2013) Heme binding induces dimerization and nitration of truncated beta-amyloid peptide Abeta16 under oxidative stress. Angew Chem Int Ed Engl 52: 8041-8044. doi: 10.1002/anie.201302989
|
| [67] |
Nathan C, Calingasan N, Nezezon J, et al. (2005) Protection from Alzheimer's-like disease in the mouse by genetic ablation of inducible nitric oxide synthase. J Exp Med 202: 1163-1169. doi: 10.1084/jem.20051529
|
| [68] | Aboud O, Parcon PA, DeWall KM, et al. (2015) Aging, Alzheimer's, and APOE genotype influence the expression and neuronal distribution patterns of microtubule motor protein dynactin-P50. Front Cell Neurosci 9: 103. |
| [69] |
Hickman SE, Allison EK, El Khoury J (2008) Microglial dysfunction and defective beta-amyloid clearance pathways in aging Alzheimer's disease mice. J Neurosci 28: 8354-8360. doi: 10.1523/JNEUROSCI.0616-08.2008
|
| [70] |
Lai AY, McLaurin J (2012) Clearance of amyloid-beta peptides by microglia and macrophages: the issue of what, when and where. Future Neurol 7: 165-176. doi: 10.2217/fnl.12.6
|
| [71] |
Hawkes CA, McLaurin J (2009) Selective targeting of perivascular macrophages for clearance of beta-amyloid in cerebral amyloid angiopathy. Proc Natl Acad Sci U S A 106: 1261-1266. doi: 10.1073/pnas.0805453106
|
| [72] |
Mildner A, Schlevogt B, Kierdorf K, et al. (2011) Distinct and non-redundant roles of microglia and myeloid subsets in mouse models of Alzheimer's disease. J Neurosci 31: 11159-11171. doi: 10.1523/JNEUROSCI.6209-10.2011
|
| [73] |
Stalder AK, Ermini F, Bondolfi L, et al. (2005) Invasion of hematopoietic cells into the brain of amyloid precursor protein transgenic mice. J Neurosci 25: 11125-11132. doi: 10.1523/JNEUROSCI.2545-05.2005
|
| [74] |
Malm TM, Koistinaho M, Parepalo M, et al. (2005) Bone-marrow-derived cells contribute to the recruitment of microglial cells in response to beta-amyloid deposition in APP/PS1 double transgenic Alzheimer mice. Neurobiol Dis 18: 134-142 doi: 10.1016/j.nbd.2004.09.009
|
| [75] |
Fiala M, Liu PT, Espinosa-Jeffrey A, et al. (2007) Innate immunity and transcription of MGAT-III and Toll-like receptors in Alzheimer's disease patients are improved by bisdemethoxycurcumin. Proc Natl Acad Sci U S A 104: 12849-12854. doi: 10.1073/pnas.0701267104
|
| [76] |
Mildner A, Schmidt H, Nitsche M, et al. (2007) Microglia in the adult brain arise from Ly-6ChiCCR2+ monocytes only under defined host conditions. Nat Neurosci 10: 1544-1553. doi: 10.1038/nn2015
|
| [77] |
Ajami B, Bennett JL, Krieger C, et al. (2007) Local self-renewal can sustain CNS microglia maintenance and function throughout adult life. Nat Neurosci 10: 1538-1543. doi: 10.1038/nn2014
|
| [78] |
Heppner FL, Ransohoff RM, Becher B (2015) Immune attack: the role of inflammation in Alzheimer disease. Nat Rev Neurosci 16: 358-372. doi: 10.1038/nrn3880
|
| [79] |
Togo T, Akiyama H, Iseki E, et al. (2002) Occurrence of T cells in the brain of Alzheimer's disease and other neurological diseases. J Neuroimmunol 124: 83-92. doi: 10.1016/S0165-5728(01)00496-9
|
| [80] |
Town T, Tan J, Flavell RA, et al. (2005) T-cells in Alzheimer's disease. Neuromolecular Med 7: 255-264. doi: 10.1385/NMM:7:3:255
|
| [81] |
Hohsfield LA, Humpel C (2015) Migration of blood cells to beta-amyloid plaques in Alzheimer's disease. Exp Gerontol 65: 8-15. doi: 10.1016/j.exger.2015.03.002
|
| [82] | Fisher Y, Nemirovsky A, Baron R, et al. (2011) Dendritic cells regulate amyloid-beta-specific T-cell entry into the brain: the role of perivascular amyloid-beta. J Alzheimers Dis 27: 99-111. |
| [83] |
Zhang J, Ke KF, Liu Z, et al. (2013) Th17 cell-mediated neuroinflammation is involved in neurodegeneration of abeta1-42-induced Alzheimer's disease model rats. PLoS One 8: e75786. doi: 10.1371/journal.pone.0075786
|
| [84] |
Gonzalez H, Pacheco R (2014) T-cell-mediated regulation of neuroinflammation involved in neurodegenerative diseases. J Neuroinflammation 11: 201. doi: 10.1186/s12974-014-0201-8
|
| [85] |
Browne TC, McQuillan K, McManus RM, et al. (2013) IFN-gamma Production by amyloid beta-specific Th1 cells promotes microglial activation and increases plaque burden in a mouse model of Alzheimer's disease. J Immunol 190: 2241-2251. doi: 10.4049/jimmunol.1200947
|
| [86] |
Fisher Y, Strominger I, Biton S, et al. (2014) Th1 polarization of T cells injected into the cerebrospinal fluid induces brain immunosurveillance. J Immunol 192: 92-102. doi: 10.4049/jimmunol.1301707
|
| [87] |
Richartz-Salzburger E, Batra A, Stransky E, et al. (2007) Altered lymphocyte distribution in Alzheimer's disease. J Psychiatr Res 41: 174-178. doi: 10.1016/j.jpsychires.2006.01.010
|
| [88] |
Bonotis K, Krikki E, Holeva V, et al. (2008) Systemic immune aberrations in Alzheimer's disease patients. J Neuroimmunol 193: 183-187. doi: 10.1016/j.jneuroim.2007.10.020
|
| [89] | Bulati M, Buffa S, Martorana A, et al. (2015) Double negative (IgG+IgD-CD27-) B cells are increased in a cohort of moderate-severe Alzheimer's disease patients and show a pro-inflammatory trafficking receptor phenotype. J Alzheimers Dis 44: 1241-1251. |
| [90] |
Speciale L, Calabrese E, Saresella M, et al. (2007) Lymphocyte subset patterns and cytokine production in Alzheimer's disease patients. Neurobiol Aging 28: 1163-1169. doi: 10.1016/j.neurobiolaging.2006.05.020
|
| [91] |
Xiao M, Hu G (2014) Involvement of aquaporin 4 in astrocyte function and neuropsychiatric disorders. CNS Neurosci Ther 20: 385-390. doi: 10.1111/cns.12267
|
| [92] |
Grolla AA, Fakhfouri G, Balzaretti G, et al. (2013) Abeta leads to Ca(2)(+) signaling alterations and transcriptional changes in glial cells. Neurobiol Aging 34: 511-522. doi: 10.1016/j.neurobiolaging.2012.05.005
|
| [93] |
Kato S, Gondo T, Hoshii Y, et al. (1998) Confocal observation of senile plaques in Alzheimer's disease: senile plaque morphology and relationship between senile plaques and astrocytes. Pathol Int 48: 332-340. doi: 10.1111/j.1440-1827.1998.tb03915.x
|
| [94] |
Kulijewicz-Nawrot M, Verkhratsky A, Chvatal A, et al. (2012) Astrocytic cytoskeletal atrophy in the medial prefrontal cortex of a triple transgenic mouse model of Alzheimer's disease. J Anat 221: 252-262. doi: 10.1111/j.1469-7580.2012.01536.x
|
| [95] |
Heneka MT, O'Banion MK, Terwel D, et al. (2010) Neuroinflammatory processes in Alzheimer's disease. J Neural Transm 117: 919-947. doi: 10.1007/s00702-010-0438-z
|
| [96] |
Avila-Munoz E, Arias C (2014) When astrocytes become harmful: functional and inflammatory responses that contribute to Alzheimer's disease. Ageing Res Rev 18: 29-40. doi: 10.1016/j.arr.2014.07.004
|
| [97] |
Wyss-Coray T, Loike JD, Brionne TC, et al. (2003) Adult mouse astrocytes degrade amyloid-beta in vitro and in situ. Nat Med 9: 453-457. doi: 10.1038/nm838
|
| [98] |
Koistinaho M, Lin S, Wu X, et al. (2004) Apolipoprotein E promotes astrocyte colocalization and degradation of deposited amyloid-beta peptides. Nat Med 10: 719-726. doi: 10.1038/nm1058
|
| [99] | Dezsi L, Tuka B, Martos D, et al. (2015) Alzheimer's disease, astrocytes and kynurenines. Curr Alzheimer Res 12: 462-480. |
| [100] |
Chavez-Gutierrez L, Bammens L, Benilova I, et al. (2012) The mechanism of gamma-Secretase dysfunction in familial Alzheimer disease. Embo j 31: 2261-2274. doi: 10.1038/emboj.2012.79
|
| [101] |
Suh J, Choi SH, Romano DM, et al. (2013) ADAM10 missense mutations potentiate beta-amyloid accumulation by impairing prodomain chaperone function. Neuron 80: 385-401. doi: 10.1016/j.neuron.2013.08.035
|
| [102] |
Jun G, Naj AC, Beecham GW, et al. (2010) Meta-analysis confirms CR1, CLU, and PICALM as alzheimer disease risk loci and reveals interactions with APOE genotypes. Arch Neurol 67: 1473-1484. doi: 10.1001/archneurol.2010.201
|
| [103] | Jin C, Liu X, Zhang F, et al. (2013) An updated meta-analysis of the association between SORL1 variants and the risk for sporadic Alzheimer's disease. J Alzheimers Dis 37: 429-437. |
| [104] |
Li Y, Rowland C, Catanese J, et al. (2008) SORL1 variants and risk of late-onset Alzheimer's disease. Neurobiol Dis 29: 293-296. doi: 10.1016/j.nbd.2007.09.001
|
| [105] |
Xiao Q, Gil SC, Yan P, et al. (2012) Role of phosphatidylinositol clathrin assembly lymphoid-myeloid leukemia (PICALM) in intracellular amyloid precursor protein (APP) processing and amyloid plaque pathogenesis. J Biol Chem 287: 21279-21289. doi: 10.1074/jbc.M111.338376
|
| [106] | Narayan P, Orte A, Clarke RW, et al. (2012) The extracellular chaperone clusterin sequesters oligomeric forms of the amyloid-beta(1-40) peptide. Nat Struct Mol Biol 19: 79-83. |
| [107] |
Lambert JC, Heath S, Even G, et al. (2009) Genome-wide association study identifies variants at CLU and CR1 associated with Alzheimer's disease. Nat Genet 41: 1094-1099. doi: 10.1038/ng.439
|
| [108] |
Terwel D, Steffensen KR, Verghese PB, et al. (2011) Critical role of astroglial apolipoprotein E and liver X receptor-alpha expression for microglial Abeta phagocytosis. J Neurosci 31: 7049-7059. doi: 10.1523/JNEUROSCI.6546-10.2011
|
| [109] |
Jiang Q, Lee CY, Mandrekar S, et al. (2008) ApoE promotes the proteolytic degradation of Abeta. Neuron 58: 681-693. doi: 10.1016/j.neuron.2008.04.010
|
| [110] |
Maezawa I, Maeda N, Montine TJ, et al. (2006) Apolipoprotein E-specific innate immune response in astrocytes from targeted replacement mice. J Neuroinflammation 3: 10. doi: 10.1186/1742-2094-3-10
|
| [111] | Maezawa I, Nivison M, Montine KS, et al. (2006) Neurotoxicity from innate immune response is greatest with targeted replacement of E4 allele of apolipoprotein E gene and is mediated by microglial p38MAPK. FASEB J 20: 797-799 |
| [112] |
Zhao L, Lin S, Bales KR, et al. (2009) Macrophage-mediated degradation of beta-amyloid via an apolipoprotein E isoform-dependent mechanism. J Neurosci 29: 3603-3612. doi: 10.1523/JNEUROSCI.5302-08.2009
|
| [113] | Seshadri S, Fitzpatrick AL, Ikram MA, et al. (2010) Genome-wide analysis of genetic loci associated with Alzheimer disease. JAMA 303: 1832-1840 |
| [114] |
Naj AC, Jun G, Beecham GW, et al. (2011) Common variants at MS4A4/MS4A6E, CD2AP, CD33 and EPHA1 are associated with late-onset Alzheimer's disease. Nat Genet 43: 436-441. doi: 10.1038/ng.801
|
| [115] |
Lambert JC, Ibrahim-Verbaas CA, Harold D, et al. (2013) Meta-analysis of 74,046 individuals identifies 11 new susceptibility loci for Alzheimer's disease. Nat Genet 45: 1452-1458. doi: 10.1038/ng.2802
|
| [116] |
Reitz C (2014) Genetic loci associated with Alzheimer's disease. Future Neurol 9: 119-122 doi: 10.2217/fnl.14.1
|
| [117] |
Malik M, Simpson JF, Parikh I, et al. (2013) CD33 Alzheimer's risk-altering polymorphism, CD33 expression, and exon 2 splicing. J Neurosci 33: 13320-13325. doi: 10.1523/JNEUROSCI.1224-13.2013
|
| [118] |
Hollingworth P, Harold D, Sims R, et al. (2011) Common variants at ABCA7, MS4A6A/MS4A4E, EPHA1, CD33 and CD2AP are associated with Alzheimer's disease. Nat Genet 43: 429-435. doi: 10.1038/ng.803
|
| [119] |
Jiang T, Yu JT, Hu N, et al. (2014) CD33 in Alzheimer's disease. Mol Neurobiol 49: 529-535. doi: 10.1007/s12035-013-8536-1
|
| [120] |
Bradshaw EM, Chibnik LB, Keenan BT, et al. (2013) CD33 Alzheimer's disease locus: altered monocyte function and amyloid biology. Nat Neurosci 16: 848-850. doi: 10.1038/nn.3435
|
| [121] | Rohn TT (2013) The triggering receptor expressed on myeloid cells 2: ""TREM-ming"" the inflammatory component associated with Alzheimer's disease. Oxid Med Cell Longev 2013: 860959. |
| [122] | Le Ber I, De Septenville A, Guerreiro R, et al. (2014) Homozygous TREM2 mutation in a family with atypical frontotemporal dementia. Neurobiol Aging 35: 2419.e2423-2415. |
| [123] | Sasaki A, Kakita A, Yoshida K, et al. (2015) Variable expression of microglial DAP12 and TREM2 genes in Nasu-Hakola disease. Neurogenetics. In press. |
| [124] | Lu Y, Liu W, Wang X (2015) TREM2 variants and risk of Alzheimer's disease: a meta-analysis. Neurol Sci. In press. |
| [125] | Ruiz A, Dols-Icardo O, Bullido MJ, et al. (2014) Assessing the role of the TREM2 p.R47H variant as a risk factor for Alzheimer's disease and frontotemporal dementia. Neurobiol Aging 35: 444.e441-444. |
| [126] | Rajagopalan P, Hibar DP, Thompson PM (2013) TREM2 and neurodegenerative disease. N Engl J Med 369: 1565-1567. |
| [127] |
Strohmeyer R, Ramirez M, Cole GJ, et al. (2002) Association of factor H of the alternative pathway of complement with agrin and complement receptor 3 in the Alzheimer's disease brain. J Neuroimmunol 131: 135-146. doi: 10.1016/S0165-5728(02)00272-2
|
| [128] |
Karch CM, Jeng AT, Nowotny P, et al. (2012) Expression of novel Alzheimer's disease risk genes in control and Alzheimer's disease brains. PLoS One 7: e50976. doi: 10.1371/journal.pone.0050976
|
| [129] |
Brouwers N, Van Cauwenberghe C, Engelborghs S, et al. (2012) Alzheimer risk associated with a copy number variation in the complement receptor 1 increasing C3b/C4b binding sites. Mol Psychiatry 17: 223-233. doi: 10.1038/mp.2011.24
|
| [130] |
Rogers J, Li R, Mastroeni D, et al. (2006) Peripheral clearance of amyloid beta peptide by complement C3-dependent adherence to erythrocytes. Neurobiol Aging 27: 1733-1739. doi: 10.1016/j.neurobiolaging.2005.09.043
|
| [131] |
Lue LF, Brachova L, Civin WH, et al. (1996) Inflammation, A beta deposition, and neurofibrillary tangle formation as correlates of Alzheimer's disease neurodegeneration. J Neuropathol Exp Neurol 55: 1083-1088. doi: 10.1097/00005072-199655100-00008
|
| [132] | Mahmoudi R, Kisserli A, Novella JL, et al. (2015) Alzheimer's disease is associated with low density of the long CR1 isoform. Neurobiol Aging 36: 1766.e1765-1712. |
| [133] |
Harold D, Abraham R, Hollingworth P, et al. (2009) Genome-wide association study identifies variants at CLU and PICALM associated with Alzheimer's disease. Nat Genet 41: 1088-1093. doi: 10.1038/ng.440
|
| [134] |
Thambisetty M, Simmons A, Velayudhan L, et al. (2010) Association of plasma clusterin concentration with severity, pathology, and progression in Alzheimer disease. Arch Gen Psychiatry 67: 739-748. doi: 10.1001/archgenpsychiatry.2010.78
|
| [135] |
Jones SE, Jomary C (2002) Clusterin. Int J Biochem Cell Biol 34: 427-431. doi: 10.1016/S1357-2725(01)00155-8
|
| [136] | Bell RD, Sagare AP, Friedman AE, et al. (2007) Transport pathways for clearance of human Alzheimer's amyloid beta-peptide and apolipoproteins E and J in the mouse central nervous system. J Cereb Blood Flow Metab 27: 909-918. |
| [137] |
Jehle AW, Gardai SJ, Li S, et al. (2006) ATP-binding cassette transporter A7 enhances phagocytosis of apoptotic cells and associated ERK signaling in macrophages. J Cell Biol 174: 547-556. doi: 10.1083/jcb.200601030
|
| [138] |
Tanaka N, Abe-Dohmae S, Iwamoto N, et al. (2011) HMG-CoA reductase inhibitors enhance phagocytosis by upregulating ATP-binding cassette transporter A7. Atherosclerosis 217: 407-414. doi: 10.1016/j.atherosclerosis.2011.06.031
|
| [139] |
Wildsmith KR, Holley M, Savage JC, et al. (2013) Evidence for impaired amyloid beta clearance in Alzheimer's disease. Alzheimers Res Ther 5: 33. doi: 10.1186/alzrt187
|
| [140] |
Chan SL, Kim WS, Kwok JB, et al. (2008) ATP-binding cassette transporter A7 regulates processing of amyloid precursor protein in vitro. J Neurochem 106: 793-804. doi: 10.1111/j.1471-4159.2008.05433.x
|
| [141] |
Kim WS, Li H, Ruberu K, et al. (2013) Deletion of Abca7 increases cerebral amyloid-beta accumulation in the J20 mouse model of Alzheimer's disease. J Neurosci 33: 4387-4394. doi: 10.1523/JNEUROSCI.4165-12.2013
|
| [142] |
Proitsi P, Lee SH, Lunnon K, et al. (2014) Alzheimer's disease susceptibility variants in the MS4A6A gene are associated with altered levels of MS4A6A expression in blood. Neurobiol Aging 35: 279-290. doi: 10.1016/j.neurobiolaging.2013.08.002
|
| [143] |
Zuccolo J, Bau J, Childs SJ, et al. (2010) Phylogenetic analysis of the MS4A and TMEM176 gene families. PLoS One 5: e9369. doi: 10.1371/journal.pone.0009369
|
| [144] |
Cruse G, Beaven MA, Music SC, et al. (2015) The CD20 homologue MS4A4 directs trafficking of KIT toward clathrin-independent endocytosis pathways and thus regulates receptor signaling and recycling. Mol Biol Cell 26: 1711-1727. doi: 10.1091/mbc.E14-07-1221
|
| [145] |
Doyle KP, Quach LN, Sole M (2015) B-lymphocyte-mediated delayed cognitive impairment following stroke. J Neurosci 35: 2133-2145. doi: 10.1523/JNEUROSCI.4098-14.2015
|
| [146] |
Ma J, Yu JT, Tan L (2015) MS4A Cluster in Alzheimer's Disease. Mol Neurobiol 51: 1240-1248. doi: 10.1007/s12035-014-8800-z
|
| [147] | Wang HF, Tan L, Hao XK, et al. (2015) Effect of EPHA1 genetic variation on cerebrospinal fluid and neuroimaging biomarkers in healthy, mild cognitive impairment and Alzheimer's disease cohorts. J Alzheimers Dis 44: 115-123. |
| [148] |
Gerlai R (2001) Eph receptors and neural plasticity. Nat Rev Neurosci 2: 205-209. doi: 10.1038/35058582
|
| [149] | Gerlai R (2002) EphB and NMDA receptors: components of synaptic plasticity coming together. Trends Neurosci 25: 180-181. |
| [150] |
Kullander K, Klein R (2002) Mechanisms and functions of Eph and ephrin signalling. Nat Rev Mol Cell Biol 3: 475-486. doi: 10.1038/nrm856
|
| [151] |
Coulthard MG, Morgan M, Woodruff TM, et al. (2012) Eph/Ephrin signaling in injury and inflammation. Am J Pathol 181: 1493-1503. doi: 10.1016/j.ajpath.2012.06.043
|
| [152] |
Ieguchi K (2015) Eph as a target in inflammation. Endocr Metab Immune Disord Drug Targets 15: 119-128. doi: 10.2174/1871530315666150316121302
|
| [153] |
Sakamoto A, Sugamoto Y, Tokunaga Y, et al. (2011) Expression profiling of the ephrin (EFN) and Eph receptor (EPH) family of genes in atherosclerosis-related human cells. J Int Med Res 39: 522-527. doi: 10.1177/147323001103900220
|
| [154] |
Viernes DR, Choi LB, Kerr WG, et al. (2014) Discovery and development of small molecule SHIP phosphatase modulators. Med Res Rev 34: 795-824. doi: 10.1002/med.21305
|
| [155] | Gold MJ, Hughes MR, Antignano F, et al. (2015) Lineage-specific regulation of allergic airway inflammation by the lipid phosphatase Src homology 2 domain-containing inositol 5-phosphatase (SHIP-1). J Allergy Clin Immunol. In press. |
| [156] | Nowakowska BA, Obersztyn E, Szymanska K, et al. (2010) Severe mental retardation, seizures, and hypotonia due to deletions of MEF2C. Am J Med Genet B Neuropsychiatr Genet 153b: 1042-1051. |
| [157] |
Xu Z, Yoshida T, Wu L, et al. (2015) Transcription factor MEF2C suppresses endothelial cell inflammation via regulation of NF-kappaB and KLF2. J Cell Physiol 230: 1310-1320. doi: 10.1002/jcp.24870
|
| [158] | Aisen PS, Luddy A, Durner M, et al. (1998) HLA-DR4 influences glial activity in Alzheimer's disease hippocampus. J Neurol Sci 161: 66-69. |
| [159] |
Zota V, Nemirovsky A, Baron R, et al. (2009) HLA-DR alleles in amyloid beta-peptide autoimmunity: a highly immunogenic role for the DRB1*1501 allele. J Immunol 183: 3522-3530. doi: 10.4049/jimmunol.0900620
|
| [160] |
Mansouri L, Messalmani M, Klai S, et al. (2015) Association of HLA-DR/DQ polymorphism with Alzheimer's disease. Am J Med Sci 349: 334-337. doi: 10.1097/MAJ.0000000000000416
|
| [161] |
Kobrosly R, van Wijngaarden E (2010) Associations between immunologic, inflammatory, and oxidative stress markers with severity of depressive symptoms: an analysis of the 2005-2006 National Health and Nutrition Examination Survey. Neurotoxicology 31: 126-133. doi: 10.1016/j.neuro.2009.10.005
|
| [162] |
Na KS, Jung HY, Kim YK (2014) The role of pro-inflammatory cytokines in the neuroinflammation and neurogenesis of schizophrenia. Prog Neuropsychopharmacol Biol Psychiatry 48: 277-286. doi: 10.1016/j.pnpbp.2012.10.022
|
| [163] |
Heneka MT, Carson MJ, El Khoury J, et al. (2015) Neuroinflammation in Alzheimer's disease. Lancet Neurol 14: 388-405. doi: 10.1016/S1474-4422(15)70016-5
|
| 1. | Pawan Kumar Shaw, Sunil Kumar, Shaher Momani, Samir Hadid, Dynamical analysis of fractional plant disease model with curative and preventive treatments, 2022, 164, 09600779, 112705, 10.1016/j.chaos.2022.112705 | |
| 2. | Anil Kumar, Pawan Kumar Shaw, Sunil Kumar, Numerical investigation of pine wilt disease using fractal–fractional operator, 2024, 0973-1458, 10.1007/s12648-024-03298-x |
Seoyoung Yoon, Yong-Ku Kim. The role of immunity and neuroinflammation in genetic predisposition and pathogenesis of Alzheimer's disease[J]. AIMS Genetics, 2015, 2(3): 230-249. doi: 10.3934/genet.2015.3.230
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 0.10 | 0.000163 | 0.000000 | 0.000163 | 0.000285 | 0.000122 | 0.000157 | 0.000006 |
| 0.20 | 0.001845 | 0.000571 | 0.001275 | 0.002186 | 0.000340 | 0.001837 | 0.000009 |
| 0.30 | 0.007628 | 0.003801 | 0.003828 | 0.008250 | 0.000622 | 0.007617 | 0.000011 |
| 0.40 | 0.020879 | 0.012700 | 0.008179 | 0.021835 | 0.000956 | 0.020866 | 0.000013 |
| 0.50 | 0.045593 | 0.030970 | 0.014624 | 0.046927 | 0.001334 | 0.045578 | 0.000015 |
| 0.60 | 0.086305 | 0.062885 | 0.023420 | 0.088057 | 0.001752 | 0.086288 | 0.000017 |
| 0.70 | 0.148030 | 0.113230 | 0.034800 | 0.150237 | 0.002207 | 0.148012 | 0.000018 |
| 0.80 | 0.236223 | 0.187245 | 0.048978 | 0.238919 | 0.002696 | 0.236203 | 0.000020 |
| 0.90 | 0.356743 | 0.290592 | 0.066151 | 0.359959 | 0.003216 | 0.356722 | 0.000021 |
| 1.00 | 0.515830 | 0.429326 | 0.086505 | 0.519596 | 0.003766 | 0.515808 | 0.000022 |
DownLoad:
CSV
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.08650 | --- | 0.00005 | 0.00377 | --- | 0.00006 | 0.00002 | --- | 0.00008 |
| 20 | 0.04419 | 0.96891 | 0.00011 | 0.00094 | 2.00107 | 0.00024 | 0.00000 | 2.97066 | 0.00032 |
| 40 | 0.02233 | 0.98472 | 0.00029 | 0.00024 | 2.00039 | 0.00067 | 0.00000 | 2.97987 | 0.00100 |
| 80 | 0.01123 | 0.99242 | 0.00006 | 0.00034 | 2.00014 | 0.00068 | 0.00000 | 2.98607 | 0.00115 |
| 160 | 0.00563 | 0.99623 | 0.00117 | 0.00001 | 2.00005 | 0.00194 | 0.00000 | 2.99030 | 0.00284 |
| 320 | 0.00282 | 0.99812 | 0.00244 | 0.00000 | 2.00002 | 0.00469 | 0.00000 | 2.99321 | 0.00578 |
| 640 | 0.00141 | 0.99906 | 0.00490 | 0.00000 | 2.00001 | 0.01049 | 0.00000 | 2.99524 | 0.01400 |
| 1280 | 0.00071 | 0.99953 | 0.00862 | 0.00000 | 2.00000 | 0.01811 | 0.00000 | 2.99664 | 0.02656 |
| 2560 | 0.00035 | 0.99977 | 0.01213 | 0.00000 | 2.00000 | 0.02224 | 0.00000 | 2.99744 | 0.04221 |
DownLoad:
CSV
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.10 | 1.486763 | 1.486763 | 0.000000 | 1.486763 | 0.000000 | 1.486763 | 0.000000 |
| 0.20 | 1.799017 | 1.813852 | 0.014835 | 1.803337 | 0.004320 | 1.799343 | 0.000326 |
| 0.30 | 2.107699 | 2.119910 | 0.012211 | 2.113254 | 0.005555 | 2.108071 | 0.000372 |
| 0.40 | 2.430043 | 2.433687 | 0.003644 | 2.436248 | 0.006205 | 2.430428 | 0.000385 |
| 0.50 | 2.774286 | 2.765897 | 0.008389 | 2.780962 | 0.006676 | 2.774675 | 0.000389 |
| 0.60 | 3.146213 | 3.123114 | 0.023099 | 3.153299 | 0.007086 | 3.146603 | 0.000390 |
| 0.70 | 3.550803 | 3.510572 | 0.040230 | 3.558285 | 0.007482 | 3.551193 | 0.000390 |
| 0.80 | 3.992836 | 3.933086 | 0.059750 | 4.000723 | 0.007887 | 3.993226 | 0.000390 |
| 0.90 | 4.477185 | 4.395448 | 0.081737 | 4.485498 | 0.008313 | 4.477573 | 0.000389 |
| 1.00 | 5.008980 | 4.902637 | 0.106343 | 5.017751 | 0.008771 | 5.009367 | 0.000387 |
DownLoad:
CSV
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.09645 | --- | 0.00006 | 0.00715 | --- | 0.00007 | 0.00029 | --- | 0.00010 |
| 20 | 0.04997 | 0.94871 | 0.00012 | 0.00183 | 1.96664 | 0.00016 | 0.00004 | 2.89943 | 0.00022 |
| 40 | 0.02544 | 0.97401 | 0.00028 | 0.00046 | 1.99007 | 0.00031 | 0.00000 | 2.98186 | 0.00054 |
| 80 | 0.01284 | 0.98703 | 0.00033 | 0.00012 | 1.99737 | 0.00059 | 0.00000 | 3.00258 | 0.00090 |
| 160 | 0.00645 | 0.99354 | 0.00064 | 0.00003 | 1.99933 | 0.00181 | 0.00000 | 3.00457 | 0.00370 |
| 320 | 0.00323 | 0.99678 | 0.00229 | 0.00001 | 1.99983 | 0.00451 | 0.00000 | 3.00313 | 0.00711 |
| 640 | 0.00162 | 0.99839 | 0.00238 | 0.00000 | 1.99996 | 0.00505 | 0.00000 | 3.00178 | 0.01928 |
| 1280 | 0.00081 | 0.99920 | 0.00589 | 0.00000 | 1.99999 | 0.00680 | 0.00000 | 3.00095 | 0.02912 |
| 2560 | 0.00040 | 0.99960 | 0.00682 | 0.00000 | 2.00000 | 0.01662 | 0.00000 | 3.00009 | 0.04033 |
DownLoad:
CSV
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 1.00 | 1.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 1.10 | 1.395965 | 1.350000 | 0.045965 | 1.391837 | 0.004127 | 1.395743 | 0.000221 |
| 1.20 | 1.892929 | 1.783674 | 0.109255 | 1.883452 | 0.009478 | 1.892442 | 0.000488 |
| 1.30 | 2.504965 | 2.312825 | 0.192141 | 2.488830 | 0.016135 | 2.504165 | 0.000800 |
| 1.40 | 3.246745 | 2.949869 | 0.296875 | 3.222565 | 0.024180 | 3.245583 | 0.001161 |
| 1.50 | 4.133514 | 3.707821 | 0.425693 | 4.099827 | 0.033687 | 4.131942 | 0.001571 |
| 1.60 | 5.181076 | 4.600266 | 0.580810 | 5.136346 | 0.044730 | 5.179043 | 0.002033 |
| 1.70 | 6.405768 | 5.641347 | 0.764422 | 6.348392 | 0.057377 | 6.403222 | 0.002546 |
| 1.80 | 7.824449 | 6.845745 | 0.978704 | 7.752755 | 0.071694 | 7.821336 | 0.003113 |
| 1.90 | 9.454479 | 8.228665 | 1.225814 | 9.366733 | 0.087746 | 9.450744 | 0.003735 |
| 2.00 | 11.313708 | 9.805821 | 1.507887 | 11.208114 | 0.105594 | 11.309296 | 0.004413 |
DownLoad:
CSV
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 1.50789 | --- | 0.00006 | 0.10559 | --- | 0.00007 | 0.00441 | --- | 0.00011 |
| 20 | 0.80303 | 0.90901 | 0.00027 | 0.02854 | 1.88746 | 0.00045 | 0.00059 | 2.89185 | 0.00064 |
| 40 | 0.41481 | 0.95299 | 0.00046 | 0.00743 | 1.94236 | 0.00091 | 0.00008 | 2.94618 | 0.00117 |
| 80 | 0.21087 | 0.97610 | 0.00074 | 0.00189 | 1.97081 | 0.00093 | 0.00001 | 2.97319 | 0.00251 |
| 160 | 0.10632 | 0.98795 | 0.00109 | 0.00048 | 1.98531 | 0.00465 | 0.00000 | 2.98662 | 0.00889 |
| 320 | 0.05338 | 0.99395 | 0.00326 | 0.00012 | 1.99263 | 0.00564 | 0.00000 | 2.99332 | 0.00976 |
| 640 | 0.02675 | 0.99697 | 0.00729 | 0.00003 | 1.99631 | 0.01485 | 0.00000 | 2.99666 | 0.02247 |
| 1280 | 0.01339 | 0.99848 | 0.01173 | 0.00001 | 1.99815 | 0.03425 | 0.00000 | 2.99833 | 0.03725 |
| 2560 | 0.00670 | 0.99924 | 0.02115 | 0.00000 | 1.99908 | 0.03590 | 0.00000 | 2.99907 | 0.04107 |
DownLoad:
CSV
| EM | IEM | RCM | ||||
| $ x $ | $ u_{EM} $ | $ \left|u_{EM}(h)-u_{EM}(\frac{h}{2})\right| $ | $ u_{IEM} $ | $ \left|u_{IEM}(h)-u_{IEM}(\frac{h}{2})\right| $ | $ u_{{RCM}} $ | $ \left|u_{RCM}(h)-u_{RCM}(\frac{h}{2})\right| $ |
| 0.00 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 0.10 | 1.196419 | 0.005035 | 1.206748 | 0.000129 | 1.206575 | 0.000001 |
| 0.20 | 1.436327 | 0.011185 | 1.459271 | 0.000287 | 1.458887 | 0.000001 |
| 0.30 | 1.729350 | 0.018697 | 1.767703 | 0.000479 | 1.767061 | 0.000002 |
| 0.40 | 2.087249 | 0.027872 | 2.144423 | 0.000715 | 2.143466 | 0.000003 |
| 0.50 | 2.524389 | 0.039079 | 2.604549 | 0.001002 | 2.603208 | 0.000005 |
| 0.60 | 3.058312 | 0.052766 | 3.166549 | 0.001353 | 3.164738 | 0.000007 |
| 0.70 | 3.710447 | 0.069484 | 3.852977 | 0.001781 | 3.850593 | 0.000009 |
| 0.80 | 4.506967 | 0.089903 | 4.691383 | 0.002305 | 4.688297 | 0.000011 |
| 0.90 | 5.479839 | 0.114843 | 5.715413 | 0.002944 | 5.711471 | 0.000014 |
| 1.00 | 6.668107 | 0.145305 | 6.966167 | 0.003725 | 6.961180 | 0.000018 |
DownLoad:
CSV
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.28317 | --- | 0.00003 | 0.01489 | --- | 0.00004 | 0.00014 | --- | 0.00005 |
| 20 | 0.14531 | 0.96258 | 0.00007 | 0.00372 | 1.99910 | 0.00007 | 0.00002 | 2.99292 | 0.00010 |
| 40 | 0.07358 | 0.98163 | 0.00011 | 0.00093 | 1.99977 | 0.00012 | 0.00000 | 2.99678 | 0.00019 |
| 80 | 0.03702 | 0.99090 | 0.00017 | 0.00023 | 1.99994 | 0.00024 | 0.00000 | 2.99847 | 0.00065 |
| 160 | 0.01857 | 0.99547 | 0.00028 | 0.00006 | 1.99999 | 0.00049 | 0.00000 | 2.99926 | 0.00075 |
| 320 | 0.00930 | 0.99774 | 0.00034 | 0.00001 | 2.00000 | 0.00063 | 0.00000 | 2.99963 | 0.00096 |
| 640 | 0.00465 | 0.99887 | 0.00104 | 0.00000 | 2.00000 | 0.00164 | 0.00000 | 2.99984 | 0.00190 |
| 1280 | 0.00233 | 0.99944 | 0.00129 | 0.00000 | 2.00000 | 0.00292 | 0.00000 | 2.99947 | 0.00471 |
| 2560 | 0.00116 | 0.99972 | 0.00334 | 0.00000 | 2.00000 | 0.00724 | 0.00000 | 3.00440 | 0.01124 |
DownLoad:
CSV
| EM | IEM | RCM | ||||||
| Year(t) | $ \mathcal{N}_{clasical} $ | $ \mathcal{N}_{frac} $ | $ \mathcal{N}_{EM} $ | $ Error $ | $ \mathcal{N}_{IEM} $ | $ Error $ | $ \mathcal{N}_{{RCM}} $ | $ Error $ |
| 1920 | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ |
| 1930 | 2.1289$ \times 10^3 $ | 1.9909$ \times 10^3 $ | 1.9799$ \times 10^3 $ | 1.1060$ \times 10^1 $ | 1.9892$ \times 10^3 $ | 1.7315$ \times 10^0 $ | 1.9904$ \times 10^3 $ | 4.8870$ \times 10^{-1} $ |
| 1940 | 2.4366$ \times 10^3 $ | 2.2169$ \times 10^3 $ | 2.2020$ \times 10^3 $ | 1.4921$ \times 10^1 $ | 2.2152$ \times 10^3 $ | 1.7413$ \times 10^0 $ | 2.2164$ \times 10^3 $ | 4.8880$ \times 10^{-1} $ |
| 1950 | 2.7888$ \times 10^3 $ | 2.5173$ \times 10^3 $ | 2.4987$ \times 10^3 $ | 1.8625$ \times 10^1 $ | 2.5156$ \times 10^3 $ | 1.7390$ \times 10^0 $ | 2.5168$ \times 10^3 $ | 4.8882$ \times 10^{-1} $ |
| 1960 | 3.1919$ \times 10^3 $ | 2.8943$ \times 10^3 $ | 2.8717$ \times 10^3 $ | 2.2615$ \times 10^1 $ | 2.8926$ \times 10^3 $ | 1.7320$ \times 10^0 $ | 2.8938$ \times 10^3 $ | 4.8884$ \times 10^{-1} $ |
| 1970 | 3.6533$ \times 10^3 $ | 3.3561$ \times 10^3 $ | 3.3290$ \times 10^3 $ | 2.7117$ \times 10^1 $ | 3.3544$ \times 10^3 $ | 1.7219$ \times 10^0 $ | 3.3556$ \times 10^3 $ | 4.8885$ \times 10^{-1} $ |
| 1980 | 4.1814$ \times 10^3 $ | 3.9147$ \times 10^3 $ | 3.8824$ \times 10^3 $ | 3.2308$ \times 10^1 $ | 3.9130$ \times 10^3 $ | 1.7090$ \times 10^0 $ | 3.9142$ \times 10^3 $ | 4.8886$ \times 10^{-1} $ |
| 1990 | 4.7858$ \times 10^3 $ | 4.5858$ \times 10^3 $ | 4.5474$ \times 10^3 $ | 3.8363$ \times 10^1 $ | 4.5841$ \times 10^3 $ | 1.6930$ \times 10^0 $ | 4.5853$ \times 10^3 $ | 4.8888$ \times 10^{-1} $ |
| 2000 | 5.4775$ \times 10^3 $ | 5.3885$ \times 10^3 $ | 5.3431$ \times 10^3 $ | 4.5471$ \times 10^1 $ | 5.3869$ \times 10^3 $ | 1.6737$ \times 10^0 $ | 5.3881$ \times 10^3 $ | 4.8889$ \times 10^{-1} $ |
| 2010 | 6.2693$ \times 10^3 $ | 6.3462$ \times 10^3 $ | 6.2923$ \times 10^3 $ | 5.3847$ \times 10^1 $ | 6.3445$ \times 10^3 $ | 1.6505$ \times 10^0 $ | 6.3457$ \times 10^3 $ | 4.8891$ \times 10^{-1} $ |
| 2020 | 7.1755$ \times 10^3 $ | 7.4865$ \times 10^3 $ | 7.4228$ \times 10^3 $ | 6.3739$ \times 10^1 $ | 7.4849$ \times 10^3 $ | 1.6229$ \times 10^0 $ | 7.4860$ \times 10^3 $ | 4.8893$ \times 10^{-1} $ |
DownLoad:
CSV
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 0.10 | 0.000163 | 0.000000 | 0.000163 | 0.000285 | 0.000122 | 0.000157 | 0.000006 |
| 0.20 | 0.001845 | 0.000571 | 0.001275 | 0.002186 | 0.000340 | 0.001837 | 0.000009 |
| 0.30 | 0.007628 | 0.003801 | 0.003828 | 0.008250 | 0.000622 | 0.007617 | 0.000011 |
| 0.40 | 0.020879 | 0.012700 | 0.008179 | 0.021835 | 0.000956 | 0.020866 | 0.000013 |
| 0.50 | 0.045593 | 0.030970 | 0.014624 | 0.046927 | 0.001334 | 0.045578 | 0.000015 |
| 0.60 | 0.086305 | 0.062885 | 0.023420 | 0.088057 | 0.001752 | 0.086288 | 0.000017 |
| 0.70 | 0.148030 | 0.113230 | 0.034800 | 0.150237 | 0.002207 | 0.148012 | 0.000018 |
| 0.80 | 0.236223 | 0.187245 | 0.048978 | 0.238919 | 0.002696 | 0.236203 | 0.000020 |
| 0.90 | 0.356743 | 0.290592 | 0.066151 | 0.359959 | 0.003216 | 0.356722 | 0.000021 |
| 1.00 | 0.515830 | 0.429326 | 0.086505 | 0.519596 | 0.003766 | 0.515808 | 0.000022 |
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.08650 | --- | 0.00005 | 0.00377 | --- | 0.00006 | 0.00002 | --- | 0.00008 |
| 20 | 0.04419 | 0.96891 | 0.00011 | 0.00094 | 2.00107 | 0.00024 | 0.00000 | 2.97066 | 0.00032 |
| 40 | 0.02233 | 0.98472 | 0.00029 | 0.00024 | 2.00039 | 0.00067 | 0.00000 | 2.97987 | 0.00100 |
| 80 | 0.01123 | 0.99242 | 0.00006 | 0.00034 | 2.00014 | 0.00068 | 0.00000 | 2.98607 | 0.00115 |
| 160 | 0.00563 | 0.99623 | 0.00117 | 0.00001 | 2.00005 | 0.00194 | 0.00000 | 2.99030 | 0.00284 |
| 320 | 0.00282 | 0.99812 | 0.00244 | 0.00000 | 2.00002 | 0.00469 | 0.00000 | 2.99321 | 0.00578 |
| 640 | 0.00141 | 0.99906 | 0.00490 | 0.00000 | 2.00001 | 0.01049 | 0.00000 | 2.99524 | 0.01400 |
| 1280 | 0.00071 | 0.99953 | 0.00862 | 0.00000 | 2.00000 | 0.01811 | 0.00000 | 2.99664 | 0.02656 |
| 2560 | 0.00035 | 0.99977 | 0.01213 | 0.00000 | 2.00000 | 0.02224 | 0.00000 | 2.99744 | 0.04221 |
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 0.10 | 1.486763 | 1.486763 | 0.000000 | 1.486763 | 0.000000 | 1.486763 | 0.000000 |
| 0.20 | 1.799017 | 1.813852 | 0.014835 | 1.803337 | 0.004320 | 1.799343 | 0.000326 |
| 0.30 | 2.107699 | 2.119910 | 0.012211 | 2.113254 | 0.005555 | 2.108071 | 0.000372 |
| 0.40 | 2.430043 | 2.433687 | 0.003644 | 2.436248 | 0.006205 | 2.430428 | 0.000385 |
| 0.50 | 2.774286 | 2.765897 | 0.008389 | 2.780962 | 0.006676 | 2.774675 | 0.000389 |
| 0.60 | 3.146213 | 3.123114 | 0.023099 | 3.153299 | 0.007086 | 3.146603 | 0.000390 |
| 0.70 | 3.550803 | 3.510572 | 0.040230 | 3.558285 | 0.007482 | 3.551193 | 0.000390 |
| 0.80 | 3.992836 | 3.933086 | 0.059750 | 4.000723 | 0.007887 | 3.993226 | 0.000390 |
| 0.90 | 4.477185 | 4.395448 | 0.081737 | 4.485498 | 0.008313 | 4.477573 | 0.000389 |
| 1.00 | 5.008980 | 4.902637 | 0.106343 | 5.017751 | 0.008771 | 5.009367 | 0.000387 |
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.09645 | --- | 0.00006 | 0.00715 | --- | 0.00007 | 0.00029 | --- | 0.00010 |
| 20 | 0.04997 | 0.94871 | 0.00012 | 0.00183 | 1.96664 | 0.00016 | 0.00004 | 2.89943 | 0.00022 |
| 40 | 0.02544 | 0.97401 | 0.00028 | 0.00046 | 1.99007 | 0.00031 | 0.00000 | 2.98186 | 0.00054 |
| 80 | 0.01284 | 0.98703 | 0.00033 | 0.00012 | 1.99737 | 0.00059 | 0.00000 | 3.00258 | 0.00090 |
| 160 | 0.00645 | 0.99354 | 0.00064 | 0.00003 | 1.99933 | 0.00181 | 0.00000 | 3.00457 | 0.00370 |
| 320 | 0.00323 | 0.99678 | 0.00229 | 0.00001 | 1.99983 | 0.00451 | 0.00000 | 3.00313 | 0.00711 |
| 640 | 0.00162 | 0.99839 | 0.00238 | 0.00000 | 1.99996 | 0.00505 | 0.00000 | 3.00178 | 0.01928 |
| 1280 | 0.00081 | 0.99920 | 0.00589 | 0.00000 | 1.99999 | 0.00680 | 0.00000 | 3.00095 | 0.02912 |
| 2560 | 0.00040 | 0.99960 | 0.00682 | 0.00000 | 2.00000 | 0.01662 | 0.00000 | 3.00009 | 0.04033 |
| EM | IEM | RCM | |||||
| $ x $ | $ u_{exact} $ | $ u_{EM} $ | $ \left| u_{exact}-u_{EM}\right| $ | $ u_{IEM} $ | $ \left|u_{exact}-u_{IEM}\right| $ | $ u_{{RCM}} $ | $ \left|u_{exact}-u_{RCM}\right| $ |
| 1.00 | 1.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 1.10 | 1.395965 | 1.350000 | 0.045965 | 1.391837 | 0.004127 | 1.395743 | 0.000221 |
| 1.20 | 1.892929 | 1.783674 | 0.109255 | 1.883452 | 0.009478 | 1.892442 | 0.000488 |
| 1.30 | 2.504965 | 2.312825 | 0.192141 | 2.488830 | 0.016135 | 2.504165 | 0.000800 |
| 1.40 | 3.246745 | 2.949869 | 0.296875 | 3.222565 | 0.024180 | 3.245583 | 0.001161 |
| 1.50 | 4.133514 | 3.707821 | 0.425693 | 4.099827 | 0.033687 | 4.131942 | 0.001571 |
| 1.60 | 5.181076 | 4.600266 | 0.580810 | 5.136346 | 0.044730 | 5.179043 | 0.002033 |
| 1.70 | 6.405768 | 5.641347 | 0.764422 | 6.348392 | 0.057377 | 6.403222 | 0.002546 |
| 1.80 | 7.824449 | 6.845745 | 0.978704 | 7.752755 | 0.071694 | 7.821336 | 0.003113 |
| 1.90 | 9.454479 | 8.228665 | 1.225814 | 9.366733 | 0.087746 | 9.450744 | 0.003735 |
| 2.00 | 11.313708 | 9.805821 | 1.507887 | 11.208114 | 0.105594 | 11.309296 | 0.004413 |
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 1.50789 | --- | 0.00006 | 0.10559 | --- | 0.00007 | 0.00441 | --- | 0.00011 |
| 20 | 0.80303 | 0.90901 | 0.00027 | 0.02854 | 1.88746 | 0.00045 | 0.00059 | 2.89185 | 0.00064 |
| 40 | 0.41481 | 0.95299 | 0.00046 | 0.00743 | 1.94236 | 0.00091 | 0.00008 | 2.94618 | 0.00117 |
| 80 | 0.21087 | 0.97610 | 0.00074 | 0.00189 | 1.97081 | 0.00093 | 0.00001 | 2.97319 | 0.00251 |
| 160 | 0.10632 | 0.98795 | 0.00109 | 0.00048 | 1.98531 | 0.00465 | 0.00000 | 2.98662 | 0.00889 |
| 320 | 0.05338 | 0.99395 | 0.00326 | 0.00012 | 1.99263 | 0.00564 | 0.00000 | 2.99332 | 0.00976 |
| 640 | 0.02675 | 0.99697 | 0.00729 | 0.00003 | 1.99631 | 0.01485 | 0.00000 | 2.99666 | 0.02247 |
| 1280 | 0.01339 | 0.99848 | 0.01173 | 0.00001 | 1.99815 | 0.03425 | 0.00000 | 2.99833 | 0.03725 |
| 2560 | 0.00670 | 0.99924 | 0.02115 | 0.00000 | 1.99908 | 0.03590 | 0.00000 | 2.99907 | 0.04107 |
| EM | IEM | RCM | ||||
| $ x $ | $ u_{EM} $ | $ \left|u_{EM}(h)-u_{EM}(\frac{h}{2})\right| $ | $ u_{IEM} $ | $ \left|u_{IEM}(h)-u_{IEM}(\frac{h}{2})\right| $ | $ u_{{RCM}} $ | $ \left|u_{RCM}(h)-u_{RCM}(\frac{h}{2})\right| $ |
| 0.00 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 |
| 0.10 | 1.196419 | 0.005035 | 1.206748 | 0.000129 | 1.206575 | 0.000001 |
| 0.20 | 1.436327 | 0.011185 | 1.459271 | 0.000287 | 1.458887 | 0.000001 |
| 0.30 | 1.729350 | 0.018697 | 1.767703 | 0.000479 | 1.767061 | 0.000002 |
| 0.40 | 2.087249 | 0.027872 | 2.144423 | 0.000715 | 2.143466 | 0.000003 |
| 0.50 | 2.524389 | 0.039079 | 2.604549 | 0.001002 | 2.603208 | 0.000005 |
| 0.60 | 3.058312 | 0.052766 | 3.166549 | 0.001353 | 3.164738 | 0.000007 |
| 0.70 | 3.710447 | 0.069484 | 3.852977 | 0.001781 | 3.850593 | 0.000009 |
| 0.80 | 4.506967 | 0.089903 | 4.691383 | 0.002305 | 4.688297 | 0.000011 |
| 0.90 | 5.479839 | 0.114843 | 5.715413 | 0.002944 | 5.711471 | 0.000014 |
| 1.00 | 6.668107 | 0.145305 | 6.966167 | 0.003725 | 6.961180 | 0.000018 |
| EM | IEM | RCM | |||||||
| $ n $ | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) | Error | EOC | CPU(sec) |
| 10 | 0.28317 | --- | 0.00003 | 0.01489 | --- | 0.00004 | 0.00014 | --- | 0.00005 |
| 20 | 0.14531 | 0.96258 | 0.00007 | 0.00372 | 1.99910 | 0.00007 | 0.00002 | 2.99292 | 0.00010 |
| 40 | 0.07358 | 0.98163 | 0.00011 | 0.00093 | 1.99977 | 0.00012 | 0.00000 | 2.99678 | 0.00019 |
| 80 | 0.03702 | 0.99090 | 0.00017 | 0.00023 | 1.99994 | 0.00024 | 0.00000 | 2.99847 | 0.00065 |
| 160 | 0.01857 | 0.99547 | 0.00028 | 0.00006 | 1.99999 | 0.00049 | 0.00000 | 2.99926 | 0.00075 |
| 320 | 0.00930 | 0.99774 | 0.00034 | 0.00001 | 2.00000 | 0.00063 | 0.00000 | 2.99963 | 0.00096 |
| 640 | 0.00465 | 0.99887 | 0.00104 | 0.00000 | 2.00000 | 0.00164 | 0.00000 | 2.99984 | 0.00190 |
| 1280 | 0.00233 | 0.99944 | 0.00129 | 0.00000 | 2.00000 | 0.00292 | 0.00000 | 2.99947 | 0.00471 |
| 2560 | 0.00116 | 0.99972 | 0.00334 | 0.00000 | 2.00000 | 0.00724 | 0.00000 | 3.00440 | 0.01124 |
| EM | IEM | RCM | ||||||
| Year(t) | $ \mathcal{N}_{clasical} $ | $ \mathcal{N}_{frac} $ | $ \mathcal{N}_{EM} $ | $ Error $ | $ \mathcal{N}_{IEM} $ | $ Error $ | $ \mathcal{N}_{{RCM}} $ | $ Error $ |
| 1920 | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ | 1.8600$ \times 10^3 $ | 0.0000$ \times 10^0 $ |
| 1930 | 2.1289$ \times 10^3 $ | 1.9909$ \times 10^3 $ | 1.9799$ \times 10^3 $ | 1.1060$ \times 10^1 $ | 1.9892$ \times 10^3 $ | 1.7315$ \times 10^0 $ | 1.9904$ \times 10^3 $ | 4.8870$ \times 10^{-1} $ |
| 1940 | 2.4366$ \times 10^3 $ | 2.2169$ \times 10^3 $ | 2.2020$ \times 10^3 $ | 1.4921$ \times 10^1 $ | 2.2152$ \times 10^3 $ | 1.7413$ \times 10^0 $ | 2.2164$ \times 10^3 $ | 4.8880$ \times 10^{-1} $ |
| 1950 | 2.7888$ \times 10^3 $ | 2.5173$ \times 10^3 $ | 2.4987$ \times 10^3 $ | 1.8625$ \times 10^1 $ | 2.5156$ \times 10^3 $ | 1.7390$ \times 10^0 $ | 2.5168$ \times 10^3 $ | 4.8882$ \times 10^{-1} $ |
| 1960 | 3.1919$ \times 10^3 $ | 2.8943$ \times 10^3 $ | 2.8717$ \times 10^3 $ | 2.2615$ \times 10^1 $ | 2.8926$ \times 10^3 $ | 1.7320$ \times 10^0 $ | 2.8938$ \times 10^3 $ | 4.8884$ \times 10^{-1} $ |
| 1970 | 3.6533$ \times 10^3 $ | 3.3561$ \times 10^3 $ | 3.3290$ \times 10^3 $ | 2.7117$ \times 10^1 $ | 3.3544$ \times 10^3 $ | 1.7219$ \times 10^0 $ | 3.3556$ \times 10^3 $ | 4.8885$ \times 10^{-1} $ |
| 1980 | 4.1814$ \times 10^3 $ | 3.9147$ \times 10^3 $ | 3.8824$ \times 10^3 $ | 3.2308$ \times 10^1 $ | 3.9130$ \times 10^3 $ | 1.7090$ \times 10^0 $ | 3.9142$ \times 10^3 $ | 4.8886$ \times 10^{-1} $ |
| 1990 | 4.7858$ \times 10^3 $ | 4.5858$ \times 10^3 $ | 4.5474$ \times 10^3 $ | 3.8363$ \times 10^1 $ | 4.5841$ \times 10^3 $ | 1.6930$ \times 10^0 $ | 4.5853$ \times 10^3 $ | 4.8888$ \times 10^{-1} $ |
| 2000 | 5.4775$ \times 10^3 $ | 5.3885$ \times 10^3 $ | 5.3431$ \times 10^3 $ | 4.5471$ \times 10^1 $ | 5.3869$ \times 10^3 $ | 1.6737$ \times 10^0 $ | 5.3881$ \times 10^3 $ | 4.8889$ \times 10^{-1} $ |
| 2010 | 6.2693$ \times 10^3 $ | 6.3462$ \times 10^3 $ | 6.2923$ \times 10^3 $ | 5.3847$ \times 10^1 $ | 6.3445$ \times 10^3 $ | 1.6505$ \times 10^0 $ | 6.3457$ \times 10^3 $ | 4.8891$ \times 10^{-1} $ |
| 2020 | 7.1755$ \times 10^3 $ | 7.4865$ \times 10^3 $ | 7.4228$ \times 10^3 $ | 6.3739$ \times 10^1 $ | 7.4849$ \times 10^3 $ | 1.6229$ \times 10^0 $ | 7.4860$ \times 10^3 $ | 4.8893$ \times 10^{-1} $ |
