Our method | Method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 1.22946×10−16 | 1.1552×10−6 | 1.4408×10−6 |
0.5 | 2.40485×10−16 | 1.0805×10−6 | 1.4007×10−6 |
0.9 | 8.83875×10−17 | 8.1511×10−7 | 1.3682×10−6 |
This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.
Citation: Waleed Mohamed Abd-Elhameed, Ahad M. Al-Sady, Omar Mazen Alqubori, Ahmed Gamal Atta. Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials[J]. AIMS Mathematics, 2024, 9(9): 25457-25481. doi: 10.3934/math.20241243
[1] | Manal Alqhtani, Khaled M. Saad . Numerical solutions of space-fractional diffusion equations via the exponential decay kernel. AIMS Mathematics, 2022, 7(4): 6535-6549. doi: 10.3934/math.2022364 |
[2] | Shazia Sadiq, Mujeeb ur Rehman . Solution of fractional boundary value problems by $ \psi $-shifted operational matrices. AIMS Mathematics, 2022, 7(4): 6669-6693. doi: 10.3934/math.2022372 |
[3] | Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830 |
[4] | Mariam Al-Mazmumy, Maryam Ahmed Alyami, Mona Alsulami, Asrar Saleh Alsulami, Saleh S. Redhwan . An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs). AIMS Mathematics, 2024, 9(11): 30548-30571. doi: 10.3934/math.20241475 |
[5] | Sunyoung Bu . A collocation methods based on the quadratic quadrature technique for fractional differential equations. AIMS Mathematics, 2022, 7(1): 804-820. doi: 10.3934/math.2022048 |
[6] | Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi . A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151 |
[7] | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta . A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials. AIMS Mathematics, 2025, 10(1): 1201-1223. doi: 10.3934/math.2025057 |
[8] | Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed . Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations. AIMS Mathematics, 2022, 7(5): 7759-7780. doi: 10.3934/math.2022436 |
[9] | Chang Phang, Abdulnasir Isah, Yoke Teng Toh . Poly-Genocchi polynomials and its applications. AIMS Mathematics, 2021, 6(8): 8221-8238. doi: 10.3934/math.2021476 |
[10] | K. Ali Khalid, Aiman Mukheimer, A. Younis Jihad, Mohamed A. Abd El Salam, Hassen Aydi . Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations. AIMS Mathematics, 2022, 7(5): 8622-8644. doi: 10.3934/math.2022482 |
This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.
A wide range of fields rely on Chebyshev polynomials (CPs). Some CPs are famously special polynomials of Jacobi polynomials (JPs). We can extract four kinds of CPs from JPs. They were employed in many applications; see [1,2,3,4]. However, others can be considered special types of generalized ultraspherical polynomials; see [5,6]. Some contributions introduced and utilized other specific kinds of generalized ultraspherical polynomials. In the sequence of papers [7,8,9,10], the authors utilized CPs of the fifth- and sixth-kinds to treat different types of differential equations (DEs). Furthermore, the eighth-kind CPs were utilized in [11,12] to solve other types of DEs.
Several phenomena that arise in different applied sciences can be better understood by delving into fractional calculus, which studies the integration and derivatives for non-integer orders. When describing important phenomena, fractional differential equations (FDEs) are vital. There are many examples of FDEs applications; see, for instance, [13,14,15]. Because it is usually not feasible to find analytical solutions for these equations, numerical methods are relied upon. Several methods were utilized to tackle various types of FDEs. Here are some techniques used to treat several FDEs: the Adomian decomposition method [16], a finite difference scheme [17], generalized finite difference method [18], Gauss collocation method [19], the inverse Laplace transform [20], the residual power series method [21], multi-step methods [22], Haar wavelet in [23], matrix methods in [24,25,26], collocation methods in [27,28,29,30], Galerkin methods in [31,32,33], and neural networks method in [34].
Among the essential FDEs are the Rayleigh-Stokes equations. The fractional Rayleigh-Stokes equation is a mathematical model for the motion of fluids with fractional derivatives. This equation is used in many areas of study, such as non-Newtonian fluids, viscoelastic fluids, and fluid dynamics. Many contributions were devoted to investigating the types of Rayleigh-Stokes from a theoretical and numerical perspective. Theoretically, one can consult [35,36,37]. Several numerical approaches were followed to solve these equations. In [38], the authors used a finite difference method for the fractional Rayleigh-Stokes equation (FRSE). In [39], a computational method for two-dimensional FRSE is followed. The authors of [40] used a finite volume element algorithm to treat a nonlinear FRSE. In [41], a numerical method is applied to handle a type of Rayleigh-Stokes problem. Discrete Hahn polynomials treated variable-order two-dimensional FRSE in [42]. The authors of [43] numerically solved the FRSE.
The significance of spectral approaches in engineering and fluid dynamics has been better understood in recent years, and this trend is being further explored in the applied sciences [44,45,46]. In these techniques, approximations to integral and differential equations are assumed by expanding a variety of polynomials, which are frequently orthogonal. The three spectral techniques used most often are the collocation, tau, and Galerkin methods. The optimal spectral approach to solving the provided equation depends on the nature of the DE and the governing conditions that regulate it. The three spectral methods use distinct trial and test functions. In the Galerkin method, the test and trial functions are chosen so that each basis function member meets the given DE's underlying constraints; see [47,48]. The tau method is not limited to a specific set of basis functions like the Galerkin approach. This is why it solves many types of DEs; see [49]. Among the spectral methods, the collocation method is the most suitable; see, for example, [50,51].
In his seminal papers [52,53], Shen explored a new idea to apply the Galerkin method. He selected orthogonal combinations of Legendre and first-kind CPs to solve the second- and fourth-order DEs. The Galerkin approach was used to discretize the problems with their governing conditions. To address the even-order DEs, the authors of [54] employed a generalizing combination to solve even-order DEs.
This paper's main contribution and significance is the development of a new Galerkin approach for treating the FRSE. The suggested technique has the advantage that it yields accurate approximations by picking a small number of the retained modes of the selected Galerkin basis functions.
The current paper has the following structure. Section 2 presents some preliminaries and essential relations. Section 3 describes a Galerkin approach for treating FRSE. A comprehensive study on the convergence analysis is studied in Section 4. Section 5 is devoted to presenting some illustrative examples to show the efficiency and applicability of our proposed method. Section 6 reports some conclusions.
This section defines the fractional Caputo derivative and reviews some of its essential properties. Next, we gather significant characteristics of the second-kind CPs. This paper will use some orthogonal combinations of the second-kind CPs to solve the FRSE.
Definition 2.1. In Caputo's sense, the fractional-order derivative of the function ξ(s) is defined as [55]
Dαξ(s)=1Γ(p−α)∫s0(s−t)p−α−1ξ(p)(t)dt,α>0,s>0, p−1<α<p,p∈N. | (2.1) |
For Dα with p−1<α<p,p∈N, the following identities are valid:
DαC=0,C is a constant, | (2.2) |
Dαsp={0,if p∈N0andp<⌈α⌉,p!Γ(p−α+1)sp−α,if p∈N0andp≥⌈α⌉, | (2.3) |
where N={1,2,...} and N0={0,1,2,…}, and ⌈α⌉ is the ceiling function.
The shifted second-kind CPs U∗j(t) are orthogonal regarding the weight function ω(t)=√t(τ−t) in the interval [0,τ] and defined as [56,57]
U∗j(t)=j∑r=0λr,jtr,j≥0, | (2.4) |
where
λr,j=22r(−1)j+r(j+r+1)!τr(2r+1)!(j−r)!, | (2.5) |
with the following orthogonality relation [56]:
∫τ0ω(t)U∗m(t)U∗n(t)dt=qm,n, | (2.6) |
where
qm,n=πτ28{1,if m=n,0,if m≠n. | (2.7) |
{U∗m(t)}m≥0 can be generated by the recursive formula:
U∗m(t)=2(2tτ−1)U∗m−1(t)−U∗m−2(t),U∗0(t)=1,U∗1(t)=2tτ−1,m≥2. | (2.8) |
The following theorem that presents the derivatives of U∗m(t) is helpful in what follows.
Theorem 2.1. [56] For all j≥n, the following formula is valid:
DnU∗j(t)=(4τ)nj−n∑p=0 (p+j+n)even(p+1)(n)12(j−n−p)(12(j−n−p))!(12(j+n+p+2))1−nU∗p(t). | (2.9) |
The following particular formulas of (2.9) give expressions for the first- and second-order derivatives.
Corollary 2.1. The following derivative formulas are valid:
DU∗j(t)=4τj−1∑p=0 (p+j)odd(p+1)U∗p(t),j≥1, | (2.10) |
D2U∗j(t)=4τ2j−2∑p=0 (p+j)even(p+1)(j−p)(j+p+2)U∗p(t),j≥2. | (2.11) |
Proof. Special cases of Theorem 2.1.
This section is devoted to analyzing a Galerkin approach to solve the following FRSE [38,58]:
vt(x,t)−Dαt[avxx(x,t)]−bvxx(x,t)=S(x,t),0<α<1, | (3.1) |
governed by the following constraints:
v(x,0)=v0(x),0<x<ℓ, | (3.2) |
v(0,t)=v1(t),v(ℓ,t)=v2(t),0<t≤τ, | (3.3) |
where a and b are two positive constants and S(x,t) is a known smooth function.
Remark 3.1. The well-posedness and regularity of the fractional Rayleigh-Stokes problem are discussed in detail in [36].
We choose the trial functions to be
φi(x)=x(ℓ−x)U∗i(x). | (3.4) |
Due to (2.6), it can be seen that {φi(x)}i≥0 satisfies the following orthogonality relation:
∫ℓ0ˆω(x)φi(x)φj(x)dx=ai,j, | (3.5) |
where
ai,j=πℓ28{1,if i=j,0,if i≠j, | (3.6) |
and ˆω(x)=1x32(ℓ−x)32.
Theorem 3.1. The second-derivative of φi(x) can be expressed explicitly in terms of U∗j(x) as
d2φi(x)dx2=i∑j=0μj,iU∗j(x), | (3.7) |
where
μj,i=−2{j+1,ifi>j, and(i+j)even,12(i+1)(i+2),ifi=j,0,otherwise. | (3.8) |
Proof. Based on the basis functions in (3.4), we can write
d2φi(x)dx2=−2U∗i(x)+2(ℓ−2x)dU∗i(x)dx+(xℓ−x2)d2U∗i(x)dx2. | (3.9) |
Using Corollary 2.1, Eq (3.9) may be rewritten as
d2φi(x)dx2=−2U∗i(x)+8j−1∑p=0 (p+j)odd(p+1)U∗p(x)−16ℓj−1∑p=0 (p+j)odd(p+1)xU∗p(x)+4ℓj−2∑p=0 (p+j)even(p+1)(j−p)(j+p+2)xU∗p(x)−(2ℓ)2j−2∑p=0 (p+j)even(p+1)(j−p)(j+p+2)x2U∗p(x). | (3.10) |
With the aid of the recurrence relation (2.8), the following recurrence relation for U∗i(x) holds:
xU∗i(x)=ℓ4[U∗i+1(x)+2U∗i(x)+U∗i−1(x)]. | (3.11) |
Moreover, the last relation enables us to write the following relation:
x2U∗i(x)=ℓ24[4U∗i+1(x)+6U∗i(x)+4U∗i−1(x)+U∗i−2(x)+U∗i+2(x)]. | (3.12) |
If we insert relations (3.11) and (3.12) into relation (3.10), and perform some computations, then we get
d2φi(x)dx2=i∑j=0μj,iU∗j(x), | (3.13) |
where
μj,i=−2{j+1,if i>j, and (i+j)even,12(i+1)(i+2),if i=j,0,otherwise. | (3.14) |
This completes the proof.
Consider the FRSE (3.1), governed by the conditions: v(0,t)=v(ℓ,t)=0.
Now, consider the following spaces:
PM(Ω)=span{φi(x)U∗j(t):i,j=0,1,…,M},XM(Ω)={v(x,t)∈PM(Ω):v(0,t)=v(ℓ,t)=0}, | (3.15) |
where Ω=(0,ℓ)×(0,τ].
The approximate solution ˆv(x,t)∈XM(Ω) may be expressed as
ˆv(x,t)=M∑i=0M∑j=0cijφi(x)U∗j(t)=φCU∗T, | (3.16) |
where
φ=[φ0(x),φ1(x),…,φM(x)], |
U∗=[U∗0(t),U∗1(t),…,U∗M(t)], |
and C=(cij)0≤i,j≤M is the unknown matrix to be determined whose order is (M+1)×(M+1).
The residual R(x,t) of Eq (3.1) may be calculated to give
R(x,t)=ˆvt(x,t)−Dαt[aˆvxx(x,t)]−bˆvxx(x,t)−S(x,t). | (3.17) |
The philosophy in applying the Galerkin method is to find ˆv(x,t)∈XM(Ω), such that
(R(x,t),φr(x)U∗s(t))ˉω(x,t)=0,0≤r≤M,0≤s≤M−1, | (3.18) |
where ˉω(x,t)=ˆω(x)ω(t). The last equation may be rewritten as
M∑i=0M∑j=0cij(φi(x),φr(x))ˆω(x)(dU∗j(t)dt,U∗s(t))ω(t)−aM∑i=0M∑j=0cij(d2φi(x)dx2,φr(x))ˆω(x)(DαtU∗j(t),U∗s(t))ω(t)−bM∑i=0M∑j=0cij(d2φi(x)dx2,φr(x))ˆω(x)(U∗j(t),U∗s(t))ω(t)=(S(x,t),φr(x)U∗s(t))ˉω(x,t). | (3.19) |
In matrix form, Eq (3.19) can be written as
ATCB−aHTCK−bHTCQ=G, | (3.20) |
where
G=(gr,s)(M+1)×M,grs=(S(x,t),φr(x)U∗s(t))ˉω(x,t), | (3.21) |
A=(ai,r)(M+1)×(M+1),ai,r=(φi(x),φr(x))ˆω(x), | (3.22) |
B=(bj,s)(M+1)×M,bj,s=(dU∗j(t)dt,U∗s(t))ω(t), | (3.23) |
H=(hir)(M+1)×(M+1),hi,r=(d2φi(x)dx2,φr(x))ˆω(x), | (3.24) |
K=(kj,s)(M+1)×M,kj,s=(DαtU∗j(t),U∗s(t))ω(t), | (3.25) |
Q=(qj,s)(M+1)×M,qj,s=(U∗j(t),U∗s(t))ω(t). | (3.26) |
Moreover, (3.2) implies that
M∑i=0M∑j=0cijai,rU∗j(0)=(v(x,0),φr(x))ˆω(x),0≤r≤M. | (3.27) |
Now, Eq (3.20) along with (3.27) constitutes a system of algebraic equations of order (M+1)2, that may be solved using a suitable numerical procedure.
Now, the derivation of the formulas of the entries of the matrices A, B, H, K and Q are given in the following theorem.
Theorem 3.2. The following definite integral formulas are valid:
(a)∫ℓ0ˆω(x)φi(x)φr(x)dx=ai,r,(b)∫ℓ0ˆω(x)d2φi(x)dx2φr(x)dx=hi,r,(c)∫τ0ω(t)U∗j(t)U∗s(t)dt=qj,s,(d)∫τ0ω(t)dU∗j(t)dtU∗s(t)dt=bj,s,(e)∫τ0ω(t)[DαtU∗j(t)]U∗s(t)dt=kj,s, | (3.28) |
where qj,s and ai,r are given respectively in Eqs (2.6) and (3.6). Also, we have
hi,r=i∑j=0μj,iγj,r, | (3.29) |
bj,s=πτ2j−1∑p=0 (p+j+1)even(p+1)δp,s, | (3.30) |
γj,r={π(r+1),ifj≥r and(r+j)even,π(j+1),ifj<r and(r+j)even,0,otherwise, | (3.31) |
δp,s={1,ifp=s,0,ifp≠s, | (3.32) |
and
kj,s=j∑k=1π4k−1(s+1)k!τ2−α(−1)j+k+s(j+k+1)!Γ(k−α+32)(2k+1)!(j−k)!Γ(k−α+1)× 3˜F2(−s,s+2,−α+k+3232,−α+k+3|1), | (3.33) |
where 3˜F2 is the regularized hypergeometric function [59].
Proof. To find the elements hi,r: Using Theorem 3.1, one has
hi,r=∫ℓ0ˆω(x)d2φi(x)dx2φr(x)dx=i∑j=0μj,i∫ℓ0ˆω(x)U∗j(x)φr(x)dx. | (3.34) |
Now, ∫ℓ0ˆω(x)U∗j(x)φr(x)dx can be calculated to give the following result:
∫ℓ0ˆω(x)U∗j(x)φr(x)dx=γj,r, | (3.35) |
and therefore, we get the following desired result:
hi,r=i∑j=0μj,iγj,r. | (3.36) |
To find the elements bj,s: Formula (2.10) along with the orthogonality relation (2.6) helps us to write
bj,s=∫τ0ω(t)dU∗j(t)dtU∗s(t)dt=πτ2j−1∑p=0 (p+j)odd(p+1)δp,s. | (3.37) |
To find kj,s: Using property (2.3) together with (2.4), one can write
kj,s=∫τ0ω(t)[DαtU∗j(t)]U∗s(t)dt=j∑k=122kk!(−1)j+k(j+k+1)!(2k+1)!τk(j−k)!Γ(k−α+1)∫τ0U∗s(t)tk−αω(t)dt=j∑k=122kk!(−1)j+k(j+k+1)!(2k+1)!(j−k)!(k−α)!s∑n=0√π22n−1τ2−α(−1)n+sΓ(n+s+2)Γ(k+n−α+32)(2n+1)!(s−n)!Γ(k+n−α+3). | (3.38) |
If we note the following identity:
s∑n=0√π22n−1τ2−α(−1)n+s(n+s+1)!Γ(k+n−α+32)(2n+1)!(s−n)!Γ(k+n−α+3)=14π(−1)s(s+1)τ−α+2 Γ(k−α+32)3˜F2(−s,s+2,−α+k+3232,−α+k+3|1), | (3.39) |
then, we get
kj,s=j∑k=1π4k−1(s+1)k!τ2−α(−1)j+k+sΓ(j+k+2)Γ(k−α+32)Γ(2k+2)(j−k)!Γ(k−α+1)× 3˜F2(−s,s+2,−α+k+3232,−α+k+3|1). | (3.40) |
Theorem 3.2 is now proved.
Remark 3.2. The following algorithm shows our proposed Galerkin technique, which outlines the necessary steps to get the approximate solutions.
Algorithm 1 Coding algorithm for the proposed technique |
Input a,b,ℓ,τ,α,v0(x), and S(x,t). |
Step 1. Assume an approximate solution ˆv(x,t) as in (3.16). |
Step 2. Apply Galerkin method to obtain the system in (3.20) and (3.27). |
Step 4. Use Theorem 3.2 to get the elements of matrices A,B,H,K and Q. |
Step 5. Use NDsolve command to solve the system in (3.20) and (3.27) to get cij. |
Output ˆv(x,t). |
Remark 3.3. Based on the following substitution:
v(x,t):=y(x,t)+(1−xℓ)v(0,t)+xℓv(ℓ,t), | (3.41) |
the FRSE (3.1) with non-homogeneous boundary conditions will convert to homogeneous ones y(0,t)=y(ℓ,t)=0.
In this section, we study the error bound for the two cases corresponding to the 1-D and 2-D Chebyshev-weighted Sobolev spaces.
Assume the following Chebyshev-weighted Sobolev spaces:
Hα,mω(t)(I1)={u:Dα+ktu∈L2ω(t)(I1),0≤k≤m}, | (4.1) |
Ymˆω(x)(I2)={u:u(0)=u(ℓ)=0 and Dkxu∈L2ˆω(x)(I2),0≤k≤m}, | (4.2) |
where I1=(0,τ) and I2=(0,ℓ) are quipped with the inner product, norm, and semi-norm
(u,v)Hα,mω(t)=m∑k=0(Dα+ktu,Dα+ktv)L2ω(t),||u||2Hα,mω(t)=(u,u)Hα,mω(t),|u|Hα,mω(t)=||Dα+mtu||L2ω(t),(u,v)Ymˆω(x)=m∑k=0(Dkxu,Dkxv)L2ˆω(x),||u||2Ymˆω(x)=(u,u)Ymˆω(x),|u|Ymˆω(x)=||Dmxu||L2ˆω(x), | (4.3) |
where 0<α<1 and m∈N.
Also, assume the following two-dimensional Chebyshev-weighted Sobolev space:
Hr,sˉω(x,t)(Ω)={u:u(0,t)=u(ℓ,t)=0 and ∂α+p+qu∂xp∂tα+q∈L2ˉω(x,t)(Ω),r≥p≥0,s≥q≥0}, | (4.4) |
equipped with the norm and semi-norm
||u||Hr,sˉω(x,t)=(r∑p=0s∑q=0||∂α+p+qu∂xp∂tα+q||2L2ˉω(x,t))12,|u|Hr,sˉω(x,t)=||∂α+r+su∂xr∂tα+s||L2ˉω(x,t), | (4.5) |
where 0<α<1 and r,s∈N.
Lemma 4.1. [60] For n∈N, n+r>1, and n+s>1, where r,s∈R are any constants, we have
Γ(n+r)Γ(n+s)≤or,snnr−s, | (4.6) |
where
or,sn=exp(r−s2(n+s−1)+112(n+r−1)+(r−s)2n). | (4.7) |
Theorem 4.1. Suppose 0<α<1, and ˉη(t)=M∑j=0ˆηjU∗j(t) is the approximate solution of η(t)∈Hα,mω(t)(I1). Then, for 0≤k≤m≤M+1, we get
||Dα+kt(η(t)−ˆη(t))||L2ω(t)≲τm−kM−54(m−k)|η(t)|2Hα,mω(t), | (4.8) |
where A≲B indicates the existence of a constant ν such that A≤νB.
Proof. The definitions of η(t) and ˆη(t) allow us to have
||Dα+kt(η(t)−ˆη(t))||2L2ω(t)=∞∑n=M+1|ˆηn|2||Dα+ktU∗n(t)||2L2ω(t)=∞∑n=M+1|ˆηn|2||Dα+ktU∗n(t)||2L2ω(t)||Dα+mtU∗n(t)||2L2ω(t)||Dα+mtU∗n(t)||2L2ω(t)≤||Dα+ktU∗M+1(t)||2L2ω(t)||Dα+mtU∗M+1(t)||2L2ω(t)|η(t)|2Hα,mω(t). | (4.9) |
To estimate the factor ||Dα+ktU∗M+1(t)||2L2ω(t)||Dα+mtU∗M+1(t)||2L2ω(t), we first find ||Dα+ktU∗M+1(t)||2L2ω(t).
||Dα+ktU∗M+1(t)||2L2ω(t)=∫τ0Dα+ktU∗M+1(t)Dα+ktU∗M+1(t)ω(t)dt. | (4.10) |
Equation (2.3) along with (2.4) allows us to write
Dα+ktU∗M+1(t)=M+1∑r=k+1λr,M+1r!Γ(r−k−α+1)tr−k−α, | (4.11) |
and accordingly, we have
||Dα+ktU∗M+1(t)||2L2ω(t)=M+1∑r=k+1λ2r,M+1(r!)2Γ2(r−k−α+1)∫τ0t2(r−k−α)+12(τ−t)12dt=M+1∑r=k+1λ2r,M+1τ2(r−k−α+1)√π(r!)2Γ(2(r−k−α)+32)2Γ2(r−k−α+1)Γ(2(r−k−α)+3). | (4.12) |
The following inequality can be obtained after applying the Stirling formula [44]:
Γ2(r+1)Γ(2(r−k−α)+32)Γ2(r−k−α+1)Γ(2(r−k−α)+3)≲r2(k+α)(r−k)−32. | (4.13) |
By virtue of the Stirling formula [44] and Lemma 4.1, ||Dα+ktU∗M+1(t)||2L2ω(t) can be written as
||Dα+ktU∗M+1(t)||2L2ω(t)≲λ∗τ2(M−k−α+2)(M+1)2(k+α)(M−k+1)−32M+1∑r=k+11=λ∗τ2(M−k−α+2)(M+1)2(k+α)(M−k+1)−12=λ∗τ2(M−k−α+2)(Γ(M+2)Γ(M+1))2(k+α)(Γ(M−k+2)Γ(M−k+1))−12≲τ2(M−k−α+2)M2(k+α)(M−k)−12, | (4.14) |
where λ∗=max0≤r≤M+1{λ2r,M+1√π2}.
Similarly, we have
||Dα+mtU∗M+1(t)||2L2ω(t)≲τ2(M−m−α+2)M2(m+α)(M−m)−12, | (4.15) |
and accordingly, we have
||Dα+ktU∗M+1(t)||2L2ω(t)||Dα+mtU∗M+1(t)||L2ω(t)≲τ2(m−k)M2(k−m)(M−kM−m)−12=τ2(m−k)M−2(m−k)(Γ(M−k+1)Γ(M−m+1))−12≲τ2(m−k)M−52(m−k). | (4.16) |
Inserting Eq (4.16) into Eq (4.9), one gets
||Dα+kt(η(t)−ˆη(t))||2L2ω(t)≲τ2(m−k)M−52(m−k)|η(t)|2Hα,mω(t). | (4.17) |
Therefore, we get the desired result.
Theorem 4.2. Suppose ˉζ(x)=∑Mi=0ˆζiφi(x), is the approximate solution of ζM(x)∈Ymˆω(x)(I2). Then, for 0≤k≤m≤M+1, we get
||Dkx(ζ(x)−ˉζ(x))||L2ˆω(x)≲ℓm−kM−14(m−k)|ζ(x)|2Ymˆω(x). | (4.18) |
Proof. At first, based on the definitions of ζ(x) and ˉζ(x), one has
||Dkx(ζ(x)−ˉζ(x)||2L2ˆω(x)=∞∑n=M+1|ˆζn|2||Dkxφn(x)||2L2ˆω(x)=∞∑n=M+1|ˆζn|2||Dkxφn(x)||2L2ˆω(x)||Dmxφn(x)||2L2ˆω(x)||Dmxφn(x)||2L2ˆω(x)≤||DkxφM+1(x)||2L2ˆω(x)||DmxφM+1(x)||2L2ˆω(x)|ζ(x)|2Ymˆω(x). | (4.19) |
Now, we have
DkxφM+1(x)=M+1∑r=kℓλr,M+1Γ(r+2)Γ(r−k+2)xr−k+1−M+1∑r=kλr,M+1Γ(r+3)Γ(r−k+3)xr−k+2, | (4.20) |
and therefore, ||DkxφM+1(x)||2L2ˆω(x) can be written as
||DkxφM+1(x)||2L2ˆω(x)=−M+1∑r=kℓ2(r−k)λ2r,M+12√πΓ2(r+2)Γ(2(r−k+1)−12)Γ2(r−k+2)Γ(2(r−k+1)−1)+M+1∑r=kℓ2(r−k+1)λ2r,M+12√πΓ2(r+3)Γ(2(r−k+2)−12)Γ2(r−k+3)Γ(2(r−k+2)−1). | (4.21) |
The application of the Stirling formula [44] leads to
Γ2(r+2)Γ(2(r−k+1)−12)Γ2(r−k+2)Γ(2(r−k+1)−1)≲r2k(r−k)12,Γ2(r+3)Γ(2(r−k+2)−12)Γ2(r−k+3)Γ(2(r−k+2)−1)≲r2k(r−k)12, | (4.22) |
and hence, we get
||DkxφM+1(x)||2L2ˆω(x)≲ℓ2(r−k+1)M2k(M−k)32. | (4.23) |
Finally, we get the following estimation:
||DkxφM+1(x)||2L2ˆω(x)||DmxφM+1(x)||2L2ˆω(x)≲ℓ2(m−k)M−12(m−k). | (4.24) |
At the end, we get
||Dkx(ζ(x)−ˉζ(x))||L2ˆω(x)≲ℓm−kM−14(m−k)|ζ(x)|2Ymˆω(x). | (4.25) |
Theorem 4.3. Given the following assumptions: α=0, 0≤p≤r≤M+1, and the approximation to v(x,t)∈Hr,s¨ω(Ω) is ˆv(x,t). As a result, the estimation that follows is applicable:
||∂p∂xp(v(x,t)−ˆv(x,t))||L2ˉω(x,t)≲ℓr−pM−14(r−p)|v(x,t)|Hr,0ˉω(x,t). | (4.26) |
Proof. According to the definitions of v(x,t) and ˆv(x,t), one has
v(x,t)−ˆv(x,t)=M∑i=0∞∑j=M+1cijφi(x)U∗j(t)+∞∑i=M+1∞∑j=0cijφi(x)U∗j(t)≤M∑i=0∞∑j=0cijφi(x)U∗j(t)+∞∑i=M+1∞∑j=0cijφi(x)U∗j(t). | (4.27) |
Now, applying the same procedures as in Theorem 4.2, we obtain
||∂p∂xp(v(x,t)−ˆv(x,t))||L2ˉω(x,t)≲ℓr−pM−14(r−p)|v(x,t)|Hr,0ˉω(x,t). | (4.28) |
Theorem 4.4. Given the following assumptions: α=0, 0≤q≤s≤M+1, and the approximation to v(x,t)∈Hr,s¨ω(Ω) is ˆv(x,t). As a result, the estimation that follows is applicable:
||∂q∂tq(v(x,t)−ˆv(x,t))||L2ˉω(x,t)≲τs−qM−54(s−q)|v(x,t)|H0,sˉω(x,t). | (4.29) |
Theorem 4.5. Let ˆv(x,t) be the approximate solution of v(x,t)∈Hr,sˉω(x,t)(Ω), and assume that 0<α<1. Consequently, for 0≤p≤r≤M+1, and 0≤q≤s≤M+1, we obtain
||∂α+q∂tα+q[∂p∂xp(v(x,t)−ˆv(x,t))]||L2ˉω(x,t)≲τs−qℓr−pM−14[5(s−q)+r−p)]|v(x,t)|Hr,0ˉω(x,t). | (4.30) |
Proof. The proofs of Theorems 4.4 and 4.5 are similar to the proof of Theorem 4.3.
Theorem 4.6. Let R(x,t) be the residual of Eq (3.1), then ||R(x,t)||L2ˉω(x,t)→0 as M→∞.
Proof. ||R(x,t)||L2ˉω(x,t) of Eq (3.28) can be written as
||R(x,t)||L2ˉω(x,t)=||ˆvt(x,t)−Dαt[aˆvxx(x,t)]−bˆvxx(x,t)−S(x,t)||L2ˉω(x,t)≤||∂∂t(v(x,t)−ˆv(x,t))||L2ˉω(x,t)−a||∂α∂tα[∂2∂x2(v(x,t)−ˆv(x,t))]||L2ˉω(x,t)−b||∂2∂x2(v(x,t)−ˆv(x,t))||L2ˉω(x,t). | (4.31) |
Now, the application of Theorems 4.3–4.5 leads to
||R(x,t)||L2ˉω(x,t)≲τs−1M−54(s−1)|v(x,t)|H0,sˉω(x,t)−aτsℓr−2M−14[5s+r−2)]|v(x,t)|Hr,0ˉω(x,t)−bℓr−2M−14(r−2)|v(x,t)|Hr,0ˉω(x,t). | (4.32) |
Therefore, it is clear that ||R(x,t)||L2ˉω(x,t)→0 as M→∞.
This section will compare our shifted second-kind Galerkin method (SSKGM) with other methods. Three test problems will be presented in this regard.
Example 5.1. [38] Consider the following equation:
vt(x,t)−Dαt[vxx(x,t)]−vx(x,t)=S(x,t),0<α<1, | (5.1) |
where
S(x,t)=2tx(x−ℓ)[(5x2−5xℓ+ℓ2)(6Γ(3−α)t1−α+3t)−x2(x−ℓ)2], | (5.2) |
governed by (3.2) and (3.3). Problem (5.1) has the exact solution: u(x,t)=x3(ℓ−x)3t2.
In Table 1, we compare the L2 errors of the SSKGM with that obtained in [38] at ℓ=τ=1. Table 2 reports the amount of time for which a central processing unit (CPU) was used for obtaining results in Table 1. These tables show the high accuracy of our method. Figure 1 illustrates the absolute errors (AEs) at different values of α at M=4 when ℓ=τ=1. Figure 2 illustrates the AEs at different values of α at M=4 when ℓ=3, and τ=2. Figure 3 shows the AEs at different α at M=4 when ℓ=10, and τ=5.
Our method | Method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 1.22946×10−16 | 1.1552×10−6 | 1.4408×10−6 |
0.5 | 2.40485×10−16 | 1.0805×10−6 | 1.4007×10−6 |
0.9 | 8.83875×10−17 | 8.1511×10−7 | 1.3682×10−6 |
CPU time of our method | CPU time of method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 30.891 | 16.828 | 67.243 |
0.5 | 35.953 | 16.733 | 67.470 |
0.9 | 31.078 | 16.672 | 67.006 |
Example 5.2. [38] Consider the following equation:
vt(x,t)−Dαt[vx(x,t)]−vx(x,t)=S(x,t),0<α<1, | (5.3) |
where
S(x,t)=u2+sin(πx)[2π2Γ(3−α)t2−α+π2t2+2t]−t4sin2(πx), |
governed by (3.2) and (3.3). Problem (5.3) has the exact solution: u(x,t)=t2sin(πx).
Table 3 compares the L2 errors of the SSKGM with those obtained by the method in [38] at ℓ=τ=1. This table shows that our results are more accurate. Table 4 reports the CPU time used for obtaining results in Table 3. Moreover, Figure 4 sketches the AEs at different values M when α=0.7, and ℓ=τ=1. Table 5 presents the maximum AEs at α=0.8 and M=8 when ℓ=τ=1. Figure 5 sketches the AEs at different α for M=10, ℓ=3 and τ=1.
Our method | Method in [38] | ||
α | M=8 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 4.97952×10−10 | 9.1909×10−5 | 5.1027×10−5 |
0.5 | 5.85998×10−10 | 8.4317×10−5 | 4.4651×10−5 |
0.9 | 4.62473×10−10 | 6.2864×10−5 | 4.0543×10−5 |
CPU time of our method | CPU time of method in [38] | ||
α | M=8 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 119.061 | 20.095 | 70.952 |
0.5 | 118.001 | 19.991 | 71.117 |
0.9 | 121.36 | 19.908 | 71.153 |
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 7.82271×10−12 | 9.69765×10−11 | 1.48667×10−10 | 2.65085×10−10 |
0.2 | 4.24386×10−11 | 6.82703×10−11 | 2.34553×10−10 | 5.1182×10−10 |
0.3 | 4.48637×10−11 | 6.28317×10−11 | 2.69028×10−10 | 4.28024×10−10 |
0.4 | 2.57132×10−11 | 5.02944×10−11 | 1.45928×10−10 | 3.26067×10−10 |
0.5 | 6.59974×10−11 | 1.22092×10−11 | 4.38141×10−10 | 7.16844×10−10 |
0.6 | 1.11684×10−11 | 1.04731×10−10 | 1.78963×10−10 | 2.80989×10−10 |
0.7 | 4.19906×10−11 | 7.33454×10−11 | 2.70484×10−10 | 4.25895×10−10 |
0.8 | 2.97515×10−11 | 1.16961×10−10 | 2.70621×10−10 | 4.63116×10−10 |
0.9 | 1.0014×10−11 | 8.68311×10−11 | 1.43244×10−10 | 2.71797×10−10 |
Example 5.3. Consider the following equation:
vt(x,t)−Dαt[vx(x,t)]−vx(x,t)=S(x,t),0<α<1, | (5.4) |
where
S(x,t)=sin(2πx)(4π2Γ(5)Γ(5−α)t4−α+4π2t4+4t3), |
governed by (3.2) and (3.3). The exact solution of this problem is: u(x,t)=t4sin(2πx).
Table 6 presents the maximum AEs at α=0.5 and M=9 when ℓ=τ=1. Figure 6 sketches the AEs at different M and α=0.9 when ℓ=τ=1.
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 4.36556×10−8 | 6.4783×10−8 | 4.6896×10−8 | 3.07364×10−8 |
0.2 | 3.41326×10−8 | 5.29923×10−8 | 2.18292×10−8 | 6.69242×10−8 |
0.3 | 4.74795×10−8 | 5.70081×10−5 | 1.14413×10−7 | 1.69409×10−7 |
0.4 | 8.64528×10−8 | 1.11593×10−7 | 1.6302×10−7 | 1.69931×10−7 |
0.5 | 2.03915×10−8 | 2.95873×10−8 | 2.78558×10−8 | 2.04027×10−11 |
0.6 | 8.36608×10−8 | 1.07773×10−7 | 1.59124×10−7 | 1.70274×10−7 |
0.7 | 1.53354×10−8 | 1.00892×10−8 | 7.07611×10−8 | 1.68702×10−7 |
0.8 | 3.88458×10−8 | 6.00347×10−8 | 2.85788×10−8 | 6.72342×10−8 |
0.9 | 3.00922×10−8 | 4.49878×10−8 | 2.82622×10−8 | 3.06797×10−8 |
Example 5.4. [61] Consider the following equation:
vt(x,t)−Dαt[vx(x,t)]−vx(x,t)=S(x,t),0<α<1, | (5.5) |
governed by the following constraints:
v(x,0)=0,0<x<1, | (5.6) |
v(0,t)=tγ+2,v(1,t)=etγ+2,0<t≤1, | (5.7) |
where
S(x,t)=ex(tγ+1(γ−t+2)−Γ(γ+3)Γ(−α+γ+3)t−α+γ+2), |
and the exact solution of this problem is: u(x,t)=extγ+2. This problem is solved for the case γ=1. In Table 7, we compare the L2 errors of the SSKGM with that obtained in [61] at different values of α. This table shows the high accuracy of our method. Figure 7 illustrates the AE (left) and the approximate solution (right) at M=7 when α=0.6.
Our method | Method in [61] | |
α | M=7 | n=m=10 |
0.1 | 1.39197×10−11 | 2.176×10−9 |
0.3 | 1.34984×10−10 | 9.045×10−9 |
0.5 | 5.87711×10−11 | 1.516×10−8 |
0.7 | 8.43536×10−12 | 1.415×10−8 |
0.9 | 7.46064×10−12 | 5.749×10−9 |
This study presented a Galerkin algorithm technique for solving the FRSE using orthogonal combinations of the second-kind CPs. The Galerkin method converts the FRSE with its underlying conditions into a matrix system whose entries are given explicitly. A suitable algebraic algorithm may be utilized to solve such a system, and by chance, the approximate solution can be obtained. We showcased the effectiveness and precision of the algorithm through a comprehensive study of the error analysis and by presenting multiple numerical examples. We think the proposed method can be applied to other types of FDEs. As an expected future work, we aim to employ this paper's developed theoretical results and suitable spectral methods to treat some other problems.
W. M. Abd-Elhameed: Conceptualization, Methodology, Validation, Formal analysis, Funding acquisition, Investigation, Project administration, Supervision, Writing–Original draft, Writing–review & editing. A. M. Al-Sady: Methodology, Validation, Writing–Original draft; O. M. Alqubori: Methodology, Validation, Investigation; A. G. Atta: Conceptualization, Methodology, Validation, Formal analysis, Visualization, Software, Writing–Original draft, Writing–review & editing. All authors have read and agreed to the published version of the manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-23-FR-70). Therefore, the authors thank the University of Jeddah for its technical and financial support.
The authors declare that they have no competing interests.
[1] |
H. M. Ahmed, Numerical solutions for singular Lane-Emden equations using shifted Chebyshev polynomials of the first kind, Contemp. Math., 4 (2023), 132–149. https://doi.org/10.37256/cm.4120232254 doi: 10.37256/cm.4120232254
![]() |
[2] |
M. Abdelhakem, A. Ahmed, D. Baleanu, M. El-Kady, Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVP: applications to certain types of real-life problems, Comput. Appl. Math., 41 (2022), 253. https://doi.org/10.1007/s40314-022-01940-0 doi: 10.1007/s40314-022-01940-0
![]() |
[3] |
A. H. Bhrawy, M. A. Abdelkawy, F. Mallawi, An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays, Bound. Value Probl., 2015 (2015), 1–20. https://doi.org/10.1186/s13661-015-0364-y doi: 10.1186/s13661-015-0364-y
![]() |
[4] |
H. M. Ahmed, W. M. Abd-Elhameed, Spectral solutions of specific singular differential equations using a unified spectral Galerkin-collocation algorithm, J. Nonlinear Math. Phys., 31 (2024), 42. https://doi.org/10.1007/s44198-024-00194-0 doi: 10.1007/s44198-024-00194-0
![]() |
[5] |
Y. Xu, An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. Appl. Math., 29 (2002), 328–343. https://doi.org/10.1016/s0196-8858(02)00017-9 doi: 10.1016/s0196-8858(02)00017-9
![]() |
[6] |
A. Draux, M. Sadik, B. Moalla, Markov-Bernstein inequalities for generalized Gegenbauer weight, Appl. Numer. Math., 61 (2011), 1301–1321. https://doi.org/10.1016/j.apnum.2011.09.003 doi: 10.1016/j.apnum.2011.09.003
![]() |
[7] |
A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem, Math. Sci., 17 (2023), 415–429. https://doi.org/10.1007/s40096-022-00460-6 doi: 10.1007/s40096-022-00460-6
![]() |
[8] |
A. Eid, M. M. Khader, A. M. Megahed, Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo-Christov model, Open Phys., 22 (2024), 20240001. https://doi.org/10.1515/phys-2024-0001 doi: 10.1515/phys-2024-0001
![]() |
[9] |
M. Obeid, M. A. Abd El Salam, J. A. Younis, Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials, Appl. Math. Sci. Eng., 31 (2023), 2187388. https://doi.org/10.1080/27690911.2023.2187388 doi: 10.1080/27690911.2023.2187388
![]() |
[10] |
K. Sadri, H. Aminikhah, A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation, Int. J. Comput. Math., 99 (2022), 966–992. https://doi.org/10.1080/00207160.2021.1940977 doi: 10.1080/00207160.2021.1940977
![]() |
[11] |
W. M. Abd-Elhameed, Y. H. Youssri, A. K. Amin, A. G. Atta, Eighth-kind Chebyshev polynomials collocation algorithm for the nonlinear time-fractional generalized Kawahara equation, Fractal Fract., 7 (2023), 1–23. https://doi.org/10.3390/fractalfract7090652 doi: 10.3390/fractalfract7090652
![]() |
[12] |
H. M. Ahmed, R. M. Hafez, W. M. Abd-Elhameed, A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices, Phys. Scr., 99 (2024), 045250. https://doi.org/10.1088/1402-4896/ad3482 doi: 10.1088/1402-4896/ad3482
![]() |
[13] |
R. L. Magin, Fractional calculus in bioengineering, Part 1, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10 doi: 10.1615/CritRevBiomedEng.v32.i1.10
![]() |
[14] | V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Berlin, Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-14003-7 |
[15] | D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, Dordrecht: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5 |
[16] |
D. Albogami, D. Maturi, H. Alshehri, Adomian decomposition method for solving fractional time-Klein-Gordon equations using Maple, Appl. Math., 14 (2023), 411–418. https://doi.org/10.4236/am.2023.146024 doi: 10.4236/am.2023.146024
![]() |
[17] |
A. Kanwal, S. Boulaaras, R. Shafqat, B. Taufeeq, M. ur Rahman, Explicit scheme for solving variable-order time-fractional initial boundary value problems, Sci. Rep., 14 (2024), 5396. https://doi.org/10.1038/s41598-024-55943-4 doi: 10.1038/s41598-024-55943-4
![]() |
[18] |
L. Y. Qing, X. L. Li, Meshless analysis of fractional diffusion-wave equations by generalized finite difference method, Appl. Math. Lett., 157 (2024), 109204. https://doi.org/10.1016/j.aml.2024.109204 doi: 10.1016/j.aml.2024.109204
![]() |
[19] |
A. Z. Amin, A. M. Lopes, I. Hashim, A space-time spectral collocation method for solving the variable-order fractional Fokker-Planck equation, J. Appl. Anal. Comput., 13 (2023), 969–985. https://doi.org/10.11948/20220254 doi: 10.11948/20220254
![]() |
[20] |
Kamran, S. Ahmad, K. Shah, T. Abdeljawad, B. Abdalla, On the approximation of fractal-fractional differential equations using numerical inverse Laplace transform methods, Comput. Model. Eng. Sci., 135 (2023), 2743–2765. https://doi.org/10.32604/cmes.2023.023705 doi: 10.32604/cmes.2023.023705
![]() |
[21] |
A. Burqan, R. Saadeh, A. Qazza, S. Momani, ARA-residual power series method for solving partial fractional differential equations, Alexandria Eng. J., 62 (2023), 47–62. https://doi.org/10.1016/j.aej.2022.07.022 doi: 10.1016/j.aej.2022.07.022
![]() |
[22] |
S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y. M. Chu, On multi-step methods for singular fractional q-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
![]() |
[23] |
R. Amin, K. Shah, M. Asif, I. Khan, F. Ullah, An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math., 381 (2021), 113028. https://doi.org/10.1016/j.cam.2020.113028 doi: 10.1016/j.cam.2020.113028
![]() |
[24] |
H. M. Ahmed, New generalized Jacobi Galerkin operational matrices of derivatives: an algorithm for solving multi-term variable-order time-fractional diffusion-wave equations, Fractal Fract., 8 (2024), 1–26. https://doi.org/10.3390/fractalfract8010068 doi: 10.3390/fractalfract8010068
![]() |
[25] |
H. M. Ahmed, Enhanced shifted Jacobi operational matrices of derivatives: spectral algorithm for solving multiterm variable-order fractional differential equations, Bound. Value Probl., 2023 (2023), 108. https://doi.org/10.1186/s13661-023-01796-1 doi: 10.1186/s13661-023-01796-1
![]() |
[26] |
M. Izadi, Ş. Yüzbaşı, W. Adel, A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis, Math. Sci., 17 (2023), 267–284. https://doi.org/10.1007/s40096-022-00468-y doi: 10.1007/s40096-022-00468-y
![]() |
[27] |
Y. F. Wei, Y. Guo, Y. Li, A new numerical method for solving semilinear fractional differential equation, J. Appl. Math. Comput., 68 (2022), 1289–1311. https://doi.org/10.1007/s12190-021-01566-1 doi: 10.1007/s12190-021-01566-1
![]() |
[28] |
M. A. Zaky, I. G. Ameen, M. Babatin, A. Akgül, M. Hammad, A. Lopes, Non-polynomial collocation spectral scheme for systems of nonlinear Caputo-Hadamard differential equations, Fractal Fract., 8 (2024), 1–16. https://doi.org/10.3390/fractalfract8050262 doi: 10.3390/fractalfract8050262
![]() |
[29] |
M. H. Alharbi, A. F. Abu Sunayh, A. G. Atta, W. M. Abd-Elhameed, Novel approach by shifted Fibonacci polynomials for solving the fractional Burgers equation, Fractal Fract., 8 (2024), 1–22. https://doi.org/10.3390/fractalfract8070427 doi: 10.3390/fractalfract8070427
![]() |
[30] |
M. A. Abdelkawy, A. M. Lopes, M. M. Babatin, Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order, Chaos Solitons Fract., 134 (2020), 109721. https://doi.org/10.1016/j.chaos.2020.109721 doi: 10.1016/j.chaos.2020.109721
![]() |
[31] |
M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157. https://doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
![]() |
[32] |
W. M. Abd-Elhameed, M. M. Alsuyuti, Numerical treatment of multi-term fractional differential equations via new kind of generalized Chebyshev polynomials, Fractal Fract., 7 (2023), 1–22. https://doi.org/10.3390/fractalfract7010074 doi: 10.3390/fractalfract7010074
![]() |
[33] |
R. M. Hafez, M. A. Zaky, M. A. Abdelkawy, Jacobi spectral Galerkin method for distributed-order fractional Rayleigh-Stokes problem for a generalized second grade fluid, Front. Phys., 7 (2020), 240. https://doi.org/10.3389/fphy.2019.00240 doi: 10.3389/fphy.2019.00240
![]() |
[34] |
S. M. Sivalingam, P. Kumar, V. Govindaraj, A neural networks-based numerical method for the generalized Caputo-type fractional differential equations, Math. Comput. Simul., 213 (2023), 302–323. https://doi.org/10.1016/j.matcom.2023.06.012 doi: 10.1016/j.matcom.2023.06.012
![]() |
[35] |
N. H. Tuan, N. D. Phuong, T. N. Thach, New well-posedness results for stochastic delay Rayleigh-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 347–358. https://doi.org/10.3934/dcdsb.2022079 doi: 10.3934/dcdsb.2022079
![]() |
[36] |
J. N. Wang, Y. Zhou, A. Alsaedi, B. Ahmad, Well-posedness and regularity of fractional Rayleigh-Stokes problems, Z. Angew. Math. Phys., 73 (2022), 161. https://doi.org/10.1007/s00033-022-01808-7 doi: 10.1007/s00033-022-01808-7
![]() |
[37] |
L. Peng, Y. Zhou, The well-posedness results of solutions in Besov-Morrey spaces for fractional Rayleigh-Stokes equations, Qual. Theory Dyn. Syst., 23 (2024), 43. https://doi.org/10.1007/s12346-023-00897-7 doi: 10.1007/s12346-023-00897-7
![]() |
[38] |
Z. Guan, X. D. Wang, J. Ouyang, An improved finite difference/finite element method for the fractional Rayleigh-Stokes problem with a nonlinear source term, J. Appl. Math. Comput., 65 (2021), 451–479. https://doi.org/10.1007/s12190-020-01399-4 doi: 10.1007/s12190-020-01399-4
![]() |
[39] |
Ö. Oruç, An accurate computational method for two-dimensional (2D) fractional Rayleigh-Stokes problem for a heated generalized second grade fluid via linear barycentric interpolation method, Comput. Math. Appl., 118 (2022), 120–131. https://doi.org/ 10.1016/j.camwa.2022.05.012 doi: 10.1016/j.camwa.2022.05.012
![]() |
[40] |
Y. L. Zhang, Y. H. Zhou, J. M. Wu, Quadratic finite volume element schemes over triangular meshes for a nonlinear time-fractional Rayleigh-Stokes problem, Comput. Model. Eng. Sci., 127 (2021), 487–514. https://doi.org/10.32604/cmes.2021.014950 doi: 10.32604/cmes.2021.014950
![]() |
[41] |
M. Saffarian, A. Mohebbi, High order numerical method for the simulation of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative on regular and irregular domains, Eng. Comput., 39 (2023), 2851–2868. https://doi.org/10.1007/s00366-022-01647-0 doi: 10.1007/s00366-022-01647-0
![]() |
[42] |
F. Salehi, H. Saeedi, M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274–5292. https://doi.org/10.1007/s40314-018-0631-5 doi: 10.1007/s40314-018-0631-5
![]() |
[43] |
O. Nikan, A. Golbabai, J. A. T. Machado, T. Nikazad, Numerical solution of the fractional Rayleigh-Stokes model arising in a heated generalized second-grade fluid, Eng. Comput., 37 (2021), 1751–1764. https://doi.org/10.1007/s00366-019-00913-y doi: 10.1007/s00366-019-00913-y
![]() |
[44] | J. Shen, T. Tang, L. L. Wang, Spectral methods: algorithms, analysis and applications, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7 |
[45] | B. Shizgal, Spectral methods in chemistry and physics, Dordrecht: Springer, 2015. https://doi.org/10.1007/978-94-017-9454-1 |
[46] | C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Berlin, Heidelberg: Springer, 1988. https://doi.org/10.1007/978-3-642-84108-8 |
[47] |
M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, Galerkin operational approach for multi-dimensions fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 114 (2022), 106608. https://doi.org/10.1016/j.cnsns.2022.106608 doi: 10.1016/j.cnsns.2022.106608
![]() |
[48] |
M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, B. I. Bayoumi, D. Baleanu, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Methods Appl. Sci., 42 (2019), 1389–1412. https://doi.org/10.1002/mma.5431 doi: 10.1002/mma.5431
![]() |
[49] |
E. M. Abdelghany, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, A. G. Atta, A tau approach for solving time-fractional heat equation based on the shifted sixth-kind Chebyshev polynomials, Symmetry, 15 (2023), 1–17. https://doi.org/10.3390/sym15030594 doi: 10.3390/sym15030594
![]() |
[50] |
E. Magdy, W. M. Abd-Elhameed, Y. H. Youssri, G. M. Moatimid, A. G. Atta, A potent collocation approach based on shifted Gegenbauer polynomials for nonlinear time fractional Burgers' equations, Contemp. Math., 4 (2023), 647–665. https://doi.org/10.37256/cm.4420233302 doi: 10.37256/cm.4420233302
![]() |
[51] |
W. M. Abd-Elhameed, M. S. Al-Harbi, A. G. Atta, New convolved Fibonacci collocation procedure for the Fitzhugh-Nagumo non-linear equation, Nonlinear Eng., 13 (2024), 20220332. https://doi.org/10.1515/nleng-2022-0332 doi: 10.1515/nleng-2022-0332
![]() |
[52] |
J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489–1505. https://doi.org/10.1137/0915089 doi: 10.1137/0915089
![]() |
[53] |
J. Shen, Efficient spectral-Galerkin method Ⅱ. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), 74–87. https://doi.org/10.1137/0916006 doi: 10.1137/0916006
![]() |
[54] |
E. H. Doha, W. M. Abd-Elhameed, A. H. Bhrawy, New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials, Collect. Math., 64 (2013), 373–394. https://doi.org/10.1007/s13348-012-0067-y doi: 10.1007/s13348-012-0067-y
![]() |
[55] | I. Podlubny, Fractional differential equations, Elsevier, 1998. |
[56] |
W. M. Abd-Elhameed, Y. H. Youssri, Explicit shifted second-kind Chebyshev spectral treatment for fractional Riccati differential equation, Comput. Model. Eng. Sci., 121 (2019), 1029–1049. https://doi.org/10.32604/cmes.2019.08378 doi: 10.32604/cmes.2019.08378
![]() |
[57] |
A. G. Atta, Y. H. Youssri, Shifted second-kind Chebyshev spectral collocation-based technique for time-fractional KdV-Burgers' equation, Iran. J. Math. Chem., 14 (2023), 207–224. https://doi.org/ 10.22052/ijmc.2023.252824.1710 doi: 10.22052/ijmc.2023.252824.1710
![]() |
[58] |
H. Mesgarani, Y. E. Aghdam, M. Khoshkhahtinat, B. Farnam, Analysis of the numerical scheme of the one-dimensional fractional Rayleigh-Stokes model arising in a heated generalized problem, AIP Adv., 13 (2023), 085024. https://doi.org/10.1063/5.0156586 doi: 10.1063/5.0156586
![]() |
[59] | E. W. Weisstein, Regularized hypergeometric function. Available from: https://mathworld.wolfram.com/RegularizedHypergeometricFunction.html. |
[60] |
X. D. Zhao, L. L. Wang, Z. Q. Xie, Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions, SIAM J. Numer. Anal., 51 (2013), 1443–1469. https://doi.org/10.1137/12089421X doi: 10.1137/12089421X
![]() |
[61] |
M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243–2258. https://doi.org/10.1016/j.camwa.2017.12.004 doi: 10.1016/j.camwa.2017.12.004
![]() |
1. | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, On generalized Hermite polynomials, 2024, 9, 2473-6988, 32463, 10.3934/math.20241556 | |
2. | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Abdulrahman Khalid Al-Harbi, Mohammed H. Alharbi, Ahmed Gamal Atta, Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem, 2024, 32, 2688-1594, 6200, 10.3934/era.2024288 | |
3. | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta, A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials, 2024, 12, 2227-7390, 3672, 10.3390/math12233672 | |
4. | Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta, Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation, 2024, 9, 2473-6988, 34567, 10.3934/math.20241646 | |
5. | M.H. Heydari, M. Razzaghi, M. Bayram, A numerical approach for multi-dimensional ψ-Hilfer fractional nonlinear Galilei invariant advection–diffusion equations, 2025, 68, 22113797, 108067, 10.1016/j.rinp.2024.108067 | |
6. | Youssri Hassan Youssri, Waleed Mohamed Abd-Elhameed, Amr Ahmed Elmasry, Ahmed Gamal Atta, An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials, 2025, 9, 2504-3110, 78, 10.3390/fractalfract9020078 | |
7. | Minilik Ayalew, Mekash Ayalew, Mulualem Aychluh, Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method, 2025, 2661-3352, 10.1142/S2661335224500291 | |
8. | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta, A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials, 2025, 13, 2227-7390, 300, 10.3390/math13020300 | |
9. | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Ahmed Gamal Atta, A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials, 2025, 10, 2473-6988, 1201, 10.3934/math.2025057 | |
10. | M. Hosseininia, M.H. Heydari, D. Baleanu, M. Bayram, A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations, 2025, 25, 25900374, 100541, 10.1016/j.rinam.2025.100541 | |
11. | Waleed Mohamed Abd-Elhameed, Abdulrahman Khalid Al-Harbi, Omar Mazen Alqubori, Mohammed H. Alharbi, Ahmed Gamal Atta, Collocation Method for the Time-Fractional Generalized Kawahara Equation Using a Certain Lucas Polynomial Sequence, 2025, 14, 2075-1680, 114, 10.3390/axioms14020114 |
Our method | Method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 1.22946×10−16 | 1.1552×10−6 | 1.4408×10−6 |
0.5 | 2.40485×10−16 | 1.0805×10−6 | 1.4007×10−6 |
0.9 | 8.83875×10−17 | 8.1511×10−7 | 1.3682×10−6 |
Our method | Method in [38] | ||
α | M=8 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 4.97952×10−10 | 9.1909×10−5 | 5.1027×10−5 |
0.5 | 5.85998×10−10 | 8.4317×10−5 | 4.4651×10−5 |
0.9 | 4.62473×10−10 | 6.2864×10−5 | 4.0543×10−5 |
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 7.82271×10−12 | 9.69765×10−11 | 1.48667×10−10 | 2.65085×10−10 |
0.2 | 4.24386×10−11 | 6.82703×10−11 | 2.34553×10−10 | 5.1182×10−10 |
0.3 | 4.48637×10−11 | 6.28317×10−11 | 2.69028×10−10 | 4.28024×10−10 |
0.4 | 2.57132×10−11 | 5.02944×10−11 | 1.45928×10−10 | 3.26067×10−10 |
0.5 | 6.59974×10−11 | 1.22092×10−11 | 4.38141×10−10 | 7.16844×10−10 |
0.6 | 1.11684×10−11 | 1.04731×10−10 | 1.78963×10−10 | 2.80989×10−10 |
0.7 | 4.19906×10−11 | 7.33454×10−11 | 2.70484×10−10 | 4.25895×10−10 |
0.8 | 2.97515×10−11 | 1.16961×10−10 | 2.70621×10−10 | 4.63116×10−10 |
0.9 | 1.0014×10−11 | 8.68311×10−11 | 1.43244×10−10 | 2.71797×10−10 |
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 4.36556×10−8 | 6.4783×10−8 | 4.6896×10−8 | 3.07364×10−8 |
0.2 | 3.41326×10−8 | 5.29923×10−8 | 2.18292×10−8 | 6.69242×10−8 |
0.3 | 4.74795×10−8 | 5.70081×10−5 | 1.14413×10−7 | 1.69409×10−7 |
0.4 | 8.64528×10−8 | 1.11593×10−7 | 1.6302×10−7 | 1.69931×10−7 |
0.5 | 2.03915×10−8 | 2.95873×10−8 | 2.78558×10−8 | 2.04027×10−11 |
0.6 | 8.36608×10−8 | 1.07773×10−7 | 1.59124×10−7 | 1.70274×10−7 |
0.7 | 1.53354×10−8 | 1.00892×10−8 | 7.07611×10−8 | 1.68702×10−7 |
0.8 | 3.88458×10−8 | 6.00347×10−8 | 2.85788×10−8 | 6.72342×10−8 |
0.9 | 3.00922×10−8 | 4.49878×10−8 | 2.82622×10−8 | 3.06797×10−8 |
Our method | Method in [61] | |
α | M=7 | n=m=10 |
0.1 | 1.39197×10−11 | 2.176×10−9 |
0.3 | 1.34984×10−10 | 9.045×10−9 |
0.5 | 5.87711×10−11 | 1.516×10−8 |
0.7 | 8.43536×10−12 | 1.415×10−8 |
0.9 | 7.46064×10−12 | 5.749×10−9 |
Our method | Method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 1.22946×10−16 | 1.1552×10−6 | 1.4408×10−6 |
0.5 | 2.40485×10−16 | 1.0805×10−6 | 1.4007×10−6 |
0.9 | 8.83875×10−17 | 8.1511×10−7 | 1.3682×10−6 |
CPU time of our method | CPU time of method in [38] | ||
α | M=4 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 30.891 | 16.828 | 67.243 |
0.5 | 35.953 | 16.733 | 67.470 |
0.9 | 31.078 | 16.672 | 67.006 |
Our method | Method in [38] | ||
α | M=8 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 4.97952×10−10 | 9.1909×10−5 | 5.1027×10−5 |
0.5 | 5.85998×10−10 | 8.4317×10−5 | 4.4651×10−5 |
0.9 | 4.62473×10−10 | 6.2864×10−5 | 4.0543×10−5 |
CPU time of our method | CPU time of method in [38] | ||
α | M=8 | h=15000, T=1128 | T=15000, h=1128 |
0.1 | 119.061 | 20.095 | 70.952 |
0.5 | 118.001 | 19.991 | 71.117 |
0.9 | 121.36 | 19.908 | 71.153 |
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 7.82271×10−12 | 9.69765×10−11 | 1.48667×10−10 | 2.65085×10−10 |
0.2 | 4.24386×10−11 | 6.82703×10−11 | 2.34553×10−10 | 5.1182×10−10 |
0.3 | 4.48637×10−11 | 6.28317×10−11 | 2.69028×10−10 | 4.28024×10−10 |
0.4 | 2.57132×10−11 | 5.02944×10−11 | 1.45928×10−10 | 3.26067×10−10 |
0.5 | 6.59974×10−11 | 1.22092×10−11 | 4.38141×10−10 | 7.16844×10−10 |
0.6 | 1.11684×10−11 | 1.04731×10−10 | 1.78963×10−10 | 2.80989×10−10 |
0.7 | 4.19906×10−11 | 7.33454×10−11 | 2.70484×10−10 | 4.25895×10−10 |
0.8 | 2.97515×10−11 | 1.16961×10−10 | 2.70621×10−10 | 4.63116×10−10 |
0.9 | 1.0014×10−11 | 8.68311×10−11 | 1.43244×10−10 | 2.71797×10−10 |
x | t=0.2 | t=0.4 | t=0.6 | t=0.8 |
0.1 | 4.36556×10−8 | 6.4783×10−8 | 4.6896×10−8 | 3.07364×10−8 |
0.2 | 3.41326×10−8 | 5.29923×10−8 | 2.18292×10−8 | 6.69242×10−8 |
0.3 | 4.74795×10−8 | 5.70081×10−5 | 1.14413×10−7 | 1.69409×10−7 |
0.4 | 8.64528×10−8 | 1.11593×10−7 | 1.6302×10−7 | 1.69931×10−7 |
0.5 | 2.03915×10−8 | 2.95873×10−8 | 2.78558×10−8 | 2.04027×10−11 |
0.6 | 8.36608×10−8 | 1.07773×10−7 | 1.59124×10−7 | 1.70274×10−7 |
0.7 | 1.53354×10−8 | 1.00892×10−8 | 7.07611×10−8 | 1.68702×10−7 |
0.8 | 3.88458×10−8 | 6.00347×10−8 | 2.85788×10−8 | 6.72342×10−8 |
0.9 | 3.00922×10−8 | 4.49878×10−8 | 2.82622×10−8 | 3.06797×10−8 |
Our method | Method in [61] | |
α | M=7 | n=m=10 |
0.1 | 1.39197×10−11 | 2.176×10−9 |
0.3 | 1.34984×10−10 | 9.045×10−9 |
0.5 | 5.87711×10−11 | 1.516×10−8 |
0.7 | 8.43536×10−12 | 1.415×10−8 |
0.9 | 7.46064×10−12 | 5.749×10−9 |