Pediatric Orthogenomics: The Latest Trends and Controversies

Neha Sinha; Mark A. Seeley; Daniel S. Horwitz; Hemil Maniar; Andrea H. Seeley; Neha Sinha; Mark A. Seeley; Daniel S. Horwitz; Hemil Maniar; Andrea H. Seeley

doi:10.3934/medsci.2017.2.192

AIMS Medical Science

2017, Volume 4, Issue 2: 192-216. doi: 10.3934/medsci.2017.2.192

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Review

Pediatric Orthogenomics: The Latest Trends and Controversies

1.
Department of Orthopaedic Surgery, Geisinger Medical Center, Danville, PA, USA;
2.
Department of Pediatrics, Geisinger Medical Center, Danville, PA, USA

Received: 11 January 2017 Accepted: 12 May 2017 Published: 17 May 2017

The advent of molecular biology has paved way for an era of personalized medicine. Though medical disciplines such as oncology and cardiology are advanced in their use of genomics, implementation has been slower in other specialties, such as orthopaedics. Recent advances in genomic technology have shed light on the underlying genetic basis of various pediatric orthopaedic disorders. Prior understanding of the genetic makeup of a patient may help individualize care in patients with conditions including idiopathic scoliosis, congenital talipes equinovarus and congenital limb deformities. The fastpaced growth of information in orthogenomics often makes it challenging for an orthopaedic surgeon to effectively use this information for patient care. Genetic characterization of a patient will help indicate risk of progression of a condition, recurrence and/or response to a treatment modality, and a collaborative approach between an orthopaedic surgeon and a geneticist can help tailor patient care. The following review article summarizes current understanding in molecular genomics of common pediatric orthopaedic disorders.

Keywords:

Citation: Neha Sinha, Mark A. Seeley, Daniel S. Horwitz, Hemil Maniar, Andrea H. Seeley. Pediatric Orthogenomics: The Latest Trends and Controversies[J]. AIMS Medical Science, 2017, 4(2): 192-216. doi: 10.3934/medsci.2017.2.192

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Abstract

1. Introduction

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:

$\begin{equation} _{\mathcal C}\mathcal{D}^{\beta}_{x_0^+}u = g(x,u),\; \; {\rm{with\;initial\; condition}}\; u(x_0) = u_0,\; \; {\rm{and}}\; x\in (x_0, x_{{\rm{End}}}]. \end{equation}$

(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.

2. Preliminaries

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

$\begin{align*} &J_{a+}^{\beta}\kappa (\zeta) = \frac{1}{\Gamma(\beta)}\int_a^\zeta\frac{\kappa (p)}{(\zeta-p)^{1-\beta}}dp,\; \zeta > a,\\ { {\rm{and}}}\; \; &\\ &J_{b-}^{\beta}\kappa (\zeta) = \frac{1}{\Gamma(\beta)}\int_\zeta^b\frac{\kappa (p)}{(p-\zeta)^{1-\beta}}dp,\; \zeta < b, \end{align*}$

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

$\begin{align*} &{}_{\mathcal RL}\mathcal D_{a+}^{\beta}\kappa (\zeta) = \frac{1}{\Gamma(n-\beta)}\dfrac{d^n}{d\zeta^n}\int_a^\zeta(\zeta-p)^{n-\beta-1}\kappa (p)dp,\; \zeta > a,\\ { {\rm{and}}}\; \; &\\ &{}_{\mathcal RL}\mathcal D_{b-}^{\beta}\kappa (\zeta) = \frac{(-1)^n}{\Gamma(n-\beta)}\dfrac{d^n}{d\zeta^n}\int_\zeta^b(p-\zeta)^{n-\beta-1}\kappa (p)dp,\; \zeta < b, \end{align*}$

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

$\begin{align*} &{}_{\mathcal RL}\mathcal D_{a+}^{\beta}\kappa (\zeta) = \frac{1}{\Gamma(1-\beta)}\dfrac{d}{d\zeta}\int_a^\zeta(\zeta-p)^{-\beta}\kappa (p)dp,\; \zeta > a,\\ { {\rm{and}}}\; &\\ &{}_{\mathcal RL}\mathcal D_{b-}^{\beta}\kappa (\zeta) = -\frac{1}{\Gamma(1-\beta)}\dfrac{d}{d\zeta}\int_\zeta^b(p-\zeta)^{-\beta}\kappa (p)dp,\; \zeta < b, \end{align*}$

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

$\begin{align*} {}_{\mathcal C}\mathcal D_{a+}^{\beta}\kappa (\zeta)& = \frac{1}{\Gamma(n-\beta)}\int_a^\zeta(\zeta-p)^{n-\beta-1}\kappa^{(n)}(p)dp,\; \zeta > a,\\ { {\rm{and}}}\; \; \; \; \; \; \; \; \; \; \; \; &\\ {}_{\mathcal C}\mathcal D_{b-}^{\beta}\kappa(\zeta)& = \frac{(-1)^n}{\Gamma(n-\beta)}\int_\zeta^b(p-\zeta)^{n-\beta-1}\kappa^{(n)}(p)dp,\; \zeta < b, \end{align*}$

called the left and right Caputo derivative respectively, where $n = 1+[{\beta}]$ .

Particularly, if we take $0 < \beta < 1$ , then

$\begin{align*} &{}_{\mathcal C}\mathcal D_{a+}^{\beta}\kappa (\zeta) = \frac{1}{\Gamma(1-\beta)}\int_a^\zeta(\zeta-p)^{-\beta}\kappa'(p)dp,\; \zeta > a,\\ { {\rm{and}}}\; &\\ &{}_{\mathcal C}\mathcal D_{b-}^{\beta}\kappa(\zeta) = -\frac{1}{\Gamma(1-\beta)}\int_\zeta^b(p-\zeta)^{-\beta}\kappa'(p)dp,\; \zeta < b, \end{align*}$

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

$\begin{align*} &{}_C\mathcal D_{a+}^{\beta}\kappa (\zeta) = {}^{\mathcal RL}\mathcal D_{a+}^{\beta}\kappa (\zeta)-\sum\limits_{k = 0}^{n-1}\kappa^{k}(a)\frac{(\zeta-a)^{k-\beta}}{\Gamma(k-\beta+1)},\\ {{\rm{and}}}\; &\\ &{}_C\mathcal D_{b-}^{\beta}\kappa (\zeta) = {}^{\mathcal RL}\mathcal D_{b-}^{\beta}\kappa (\zeta)-\sum\limits_{k = 0}^{n-1}\kappa^{k}(b)\frac{(b-\zeta)^{k-\beta}}{\Gamma(k-\beta+1)}\; , \end{align*}$

where $n = 1+[{\beta}]$ .

Definition 2.4. ^[26] The one and two parameter Mittag-Leffler function are defined by,

$\begin{align*} &E_\beta(\zeta) = \sum\limits_{n = 0}^{\infty}\frac{\zeta^n}{\Gamma(\beta n+1)}\; ,\; \beta, \zeta\in \mathbb{C};\; Re(\beta) > 0,\\ { {\rm{and}}}\; &\\ &E_{\beta,\gamma}(\zeta) = \sum\limits_{n = 0}^{\infty}\frac{\zeta^n}{\Gamma(\beta n+\gamma)}\; ,\; \beta, \gamma,\zeta\in \mathbb{C}; \; Re(\beta), Re(\gamma) > 0, \end{align*}$

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:

$\begin{equation} _{\mathcal C}\mathcal D^{\beta}_{x_0^+}u = g(x,u),\; \; \mathit{{\rm{with\;the\;condition,}}}\; u(a) = u_0\; \; \mathit{{\rm{and}}}\; x\in(x_0, x_{End}], \end{equation}$

(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.

3. Proposed scheme

In our work, we are concerned about the approximate solution of the IVP for the linear and non-linear FDEs:

$\begin{equation} _{\mathcal C}\mathcal D^{\beta}_{x_0^+}u = g(x,u),\; \; {\rm{with\;initial\;condition}}\; u(x_0) = u_0,\; \; {\rm{and}}\; x\in (x_0, x_{{\rm{End}}}]. \end{equation}$

(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:

$\begin{equation} u' = {}_{\mathcal C}\mathcal D^{1-\beta}_{x_0^+}g(x,u),\; \; {\rm{with\;initial\;condition}}\; u(x_0) = u_0,\; \; {\rm{and}}\; x\in(x_0, x_{{\rm{End}}}]. \end{equation}$

(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.

Fractional RCM of FDE

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:

$\begin{cases} &x_{k} = x_0+kh, \; {\rm{for}}\; k = 0,1,2,\dots ,N\nonumber\\ &h = x_{k+1}-x_k, \nonumber\\ &u_{k+1} = u_k+\frac{h}{9}(2l_1+2l_2+4l_3), \\ &l_1 = {}_{\mathcal C}\mathcal D^{1-\beta}_{x_0^+}g(x,u_k)\left|_{x = x_k}\right.\nonumber,\\ &l_2 = {}_{\mathcal C}\mathcal D^{1-\beta}_{x_0^+}g(x+\frac{h}{2},u_k+\frac{h}{2}l_1)\left|_{x = x_k}\right.\nonumber,\\ &l_3 = {}_{\mathcal C}\mathcal D^{1-\beta}_{x_0^+}g(x+\frac{3h}{4},u_k+\frac{3h}{4}l_2)\left|_{x = x_k}\right.\nonumber.\\ \end{cases}$

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

$\begin{equation} \begin{split} \tau(x,u) = &\frac{2}{9}G(x,u)+\frac{1}{3}G\left(x+\frac{h}{2}, u+\frac{h}{2}G(x,u)\right) \\+&\frac{4}{9}G\left(x+\frac{3h}{4}, u+ \frac{3h}{4}G\left(x+\frac{3h}{4},u+\frac{3h}{4}G\left(x+\frac{h}{2},u+\frac{h}{2}G\left(x,u\right) \right)\right)\right), \end{split} \end{equation}$

(3.3)

also satisfies the condition of Lipschitz in the variable of $u$ .

Proof.

$\begin{align*} &\left|\tau(x,u_1)- \tau(x,u_2)\right|\leq\frac{2}{9}\left|G(x,u_1)-G(x,u_2)\right|+\frac{1}{3} \left|G\left(x+\frac{h}{2},u_1+\frac{h}{2}G(x,u_1)\right) -\right.\\ &\left.G\left(x+\frac{h}{2},u_2+\frac{h}{2}G(x,u_2)\right)\right|+\frac{4}{9} \left|G\left(x+\frac{3h}{4},u_1+\frac{3h}{4}G\left(x+\frac{h}{2},u_1+\frac{h}{2}G(x,y)\right)\right) \right. \\ &\left.-G\left(x+\frac{3h}{4},u_2+\frac{3h}{4}G\left(x+\frac{h}{2},u_2+\frac{h}{2}G(x,y)\right)\right) \right| \\ & \leq M\left|u_1- u_2 \right|+ \frac{hM^2}{2}\left|u_1-u_2 \right|+\frac{h^2M^3}{6}\left|u_1-u_2\right| \\ & = M\left(1+\frac{hM}{2!}+\frac{h^2M^2}{3!}\right)\left|u_1-u_2 \right| \\ & = L_\tau \left|u_1- u_2 \right|, \end{align*}$

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

$\begin{array}{l} \bar u_{k+1} = \left. u(x_k)+\frac{h}{9}\left[2\left.{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g(x,u_k)\right|_{x = x_k}+ 3{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g\left( x+\frac{h}{2}, u_k+\frac{h}{2}\left.{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g\left(x, u_k\right)\right|_{x = x_k}\right)\right|_{x = x_k}\right. \\ \left. \left. 4{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g\left( x+\frac{3h}{4}, u_k+\frac{3h}{4}\left.{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g\left(x+\frac{h}{2}, u_k+\frac{h}{2}\left.{}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g\left(x, u_k\right)\right|_{x = x_k}\right)\right|_{x = x_k}\right)\right|_{x = x_k}\right]. \end{array}$

Assuming, $G(x, u) = {}_{{\mathcal C}}\mathcal D^{1-\beta}_{x_0^+} g(x, u)$ , then

$\begin{array}{l} \bar u_{k+1} = u(x_k)+\frac{h}{9}\left[2G\left(x_k,u_k\right)+ 3 G\left(x_k+\frac{h}{2}, u_k+\frac{h}{2}u'(x_k)\right)+\right. \\ \qquad\qquad\qquad\qquad\qquad\qquad\left. 4G\left(x_k+\frac{3h}{4}, u_k+\frac{3h}{4}G\left(x_k+\frac{h}{2},u_k+\frac{h}{2}u'(x_k)\right)\right)\right] \\ = u(x_k)+\frac{2h}{9} G(x_k,u_k)+\frac{h}{3}\left[ G(x_k,u_k)+\left(\frac{h}{2}G_x(x_k,u_k)+\frac{h}{2}G_u(x_k,u_k)u'(x_k)\right)+ \right. \\ \; \left. \frac{1}{2!}\left(\left(\frac{h}{2}\right)^2 G_{xx}(x_k,u_k)+ \frac{h^2}{2}u'(x_k) \; G_{xu}(x_k,u_k)+ \left(\frac{h}{2}\right)^2 \left(u'(x_k)\right)^2 G_{uu}(x_k,u_k)\right)+O(h^3) \right]+\frac{4h}{9}\\ \left[G(x_k,u_k)+ \left(\frac{3h}{4}G_x(x_k,u_k)+\frac{3h}{4}\left(G(x_k,u_k)+\frac{h}{2}G_x(x_k,u_k)+\frac{h}{2}u'(x_k)G_u(x_k,u_k) \right)G_u(x_k,u_k)\right) \right. \\ \left.+\frac{1}{2!}\left(\frac{3h}{4}\right)^2\left(G_{xx}(x_k,u_k)+2G_{xu}(x_k,u_k)\left(G(x_k,u_k)+O(h)\right)+G_{uu}(x_k,u_k)\left(G(x_k,u_k)+O(h)\right)^2\right) \right. \\ \; +\frac{1}{3!}\left(\frac{3h}{4}\right)^3\left(G_{xxx}(\xi,\eta)+3\left(G(x_k,u_k)+O(h)\right) G_{xxu}(\xi,\eta)+3\left(G(x_k,u_k)+O(h)\right)^2G_{xuu}(\xi,\eta)\right. \\ \; \left.\left.+\left(G(x_k,u_k)+O(h)\right)^3G_{uuu}(\xi,\eta)\right)\right] \\ = u(x_k)+hu'(x_k)+\frac{h^2}{2}\left[G_{x}(x_k,u_k)+u'(x_k)G_{u}(x_k,u_k)\right]+\frac{h^3}{3!}\left[G_{xx}(x_k,u_k)+2u'(x_k)G_{xu}(x_k,u_k)\right.\\ \left.\; +G_{uu}(x_k,u_k)\left(u'(x_k)\right)^2+G_{x}(x_k,u_k)G_{y}(x_k,u_k)+u'(x_k)\left(G_{u}(x_k,u_k)\right)^2\right] +O(h^4)\\ = u(x_k)+hu'(x_k)+\frac{h^2}{2}u''(x_k)+\frac{h^3}{3!}u'''(x_k)+O(h^4). \end{array}$

(3.4)

By using the Taylor's series, the exact form of the solution will be:

$\begin{equation} u(x_{k+1}) = u(x_k)+hu'(x_k)+\frac{h^2}{2!}u''(x_k)+\frac{h^3}{3!}u'''(x_k)+\frac{h^4}{4!}u''''(x_k)+\dots \end{equation}$

(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,

$\begin{equation*} \begin{split} \tau = &\frac{2}{9}G(x,u)+\frac{1}{3}G\left(x+\frac{h}{2}, u+\frac{h}{2}G(x,u)\right) \\+&\frac{4}{9}G\left(x+\frac{3h}{4}, u+ \frac{3h}{4}G\left(x+\frac{3h}{4},u+\frac{3h}{4}G\left(x+\frac{h}{2},u+\frac{h}{2}G\left(x,u\right) \right)\right)\right), \end{split} \end{equation*}$

then using the stated Lemmas 3.1 and 3.2, we have

$\begin{align*} \left| \bar u_{k+1}-u_{k+1} \right|&\leq \left| u(x_k)-u_k\right|+h\left|\tau(x_k,u(x_k))- \tau(x_k, u_k) \right| \\ &\leq (1+hL_\tau)\left| u(x_k)-u_k\right|. \end{align*}$

Now,

$\begin{align*} \left| u(x_{k+1})- u_{k+1} \right|&\leq\left| u(x_{k+1})-\bar u_{k+1} \right|+\left| \bar u_{k+1}-u_{k+1} \right|\\ & \leq Kh^4+(1+hL_\tau)\left| u(x_k)-u_k\right|. \end{align*}$

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.

4. Stability of proposed scheme

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,

$\begin{equation} u' = G(x,u),\; \; {\rm{with \;initial\; condition}}\; u(x_0) = u_0,\; {\rm{and}}\; x\in[x_0, x_{{\rm{End}}}]. \end{equation}$

(4.1)

The numerical solution of the RCM scheme is given by the formula:

$\begin{cases} &u_{k+1} = u_k+\frac{h}{9}(2l_1+2l_2+4l_3), \\ &l_1 = G(x_k,u_k),\\ &l_2 = G\left(x_k+\frac{h}{2},u_k+\frac{h}{2}l_1\right),\\ &l_3 = G\left(x_k+\frac{3h}{4},u_k+\frac{3h}{4}l_2\right).\nonumber\\ \end{cases}$

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,

$\begin{align*} &l_1 = \mu u_k,\\ &l_2 = \mu\left(1+\frac{\mu h}{2}\right)u_k,\\ &l_3 = \mu\left(1+\frac{3\mu h}{4}\left(1+\frac{\mu h}{2}\right)\right)u_k. \end{align*}$

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 . 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.

Figure 1. Stability region for RCM.

		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
$x$	$u_{EM}$	$\left\|u_{EM}(h)-u_{EM}(\frac{h}{2})\right\|$	$u_{IEM}$	$\left\|u_{IEM}(h)-u_{IEM}(\frac{h}{2})\right\|$	$u_{{RCM}}$	$\left\|u_{RCM}(h)-u_{RCM}(\frac{h}{2})\right\|$
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
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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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

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Abstract

1. Introduction

2. Preliminaries

3. Proposed scheme

Fractional RCM of FDE

4. Stability of proposed scheme

5. Numerical results and discussion

Numerical examples

6. Application example and simulation

7. Conclusions

Acknowledgments

Conflict of interest

References

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