# An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

Authors: M. Bishehniasar; S. Salahshour; A. Ahmadian; F. Ismail; D. Baleanu
Journal: Complexity
 (2017)

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## Abstract

The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.


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## Body

## 1. Introduction

Many anomalous diffusion processes which existed in some physical and biological areas can be modeled by the time-fractional reaction diffusion wave equation. In the past few decades, high and rapid growing attention related to partial differential equations (PDEs) which contain fractional derivatives and integrals occurred due to their important application in modeling of many anomalous diffusion processes. Fractional partial differential equations (FPDEs) are excellent instrument, bringing into a broader paradigm concepts of science and engineering, such as fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks [1–16]. Consequently, the solution of FPDEs represents nowadays a vigorous research area for scientists and finding approximate and exact solutions to FPDEs is an important task. However, PDEs are commonly hard to tackle, and their fractional-order types are more complicated [1, 2, 17, 18].In recent years, several analytical and approximate techniques such as the Adomian decomposition [19], homotopy analysis [20], tau method [21], and variational iteration method [22, 23] have been constructed for solving FPDEs. Nevertheless, the study of PDEs with fractional derivative has been impeded because of the nonappearance of low cost and accurate techniques to deal with them. In addition, the derivation of approximate solution of FPDEs remains a hotspot and demands to endeavor some proficient and solid plans are an issue of serious interest.Based on the depiction above, we undoubtedly need to explore new schemes that propose prompt and obvious typical terms of analytic solutions and additionally numerical approximate solutions without linearization or discretization. Therefore, a sincere attempt has been made in this research to implement relatively new approximate-analytical technique for nonhomogeneous PDEs with a noninteger derivative of orderα∈(0,1] as(1)cDtαux,t-uxxx,t=gt,x∈0,L,t>0subject to the initial and boundary conditions:(2)ux,0=fx,0<x<L,u0,t=θt,uL,t=ωt,t>0.The analytical solutions of the FPDEs were investigated in literature by employing of Green’s functions or Fourier-Laplace transforms [24–26]. However, explicit analytic solutions of FPDEs are seldom obtainable in the literature due to extra complexity of dealing with fractional derivative. The aim of the current letter is to extend the application of the separation-variables method to solve the approximating PDE corresponding to the FPDE (1). In fact, we combine an algorithm based on the Laplace transform method to convert the FPDE to the relevant PDE [27] and a separation of variables scheme to achieve the analytical solution of the derived PDE which is close to the exact solution of the original FPDE.The separation of variables method, at least, in its most basic structures, for example, in Cartesian, spherical, or ellipsoidal coordinates, is a key part of the basic mathematical modules. In short, the separation of variables can be portrayed as a tool to reduce a multidimensional problem to series of one-dimensional ones. It behaves similar to most of global numerical techniques for solving complex models arising in real-world systems [28–31]. This urges us to exploit the powerful properties of the separation-variables method for solving FPDEs as well as reducing the difficulty of working with fractional derivatives.In terms of numerical approach, this strategy provides more efficient tool for computing approximate solutions of FPDEs in comparison with the most common numerical methods such as finite difference techniques. These stepping-type techniques need to manage the matter of stability. In either case, the base stride size for stability can be specified analytically; yet ordinarily the step sizes are not large. Note likewise that stability does not suggest exactness, particularly while approximating noninteger derivatives. Other numerical methods such as finite element methods [32–34] were developed for numerically solving FPDEs but again such a solution requires the discretization of domain into the number of finite domains/points and their computational difficulties increment quickly with the number of testing nodes. Besides that, rounding-off errors solemnly influence the solution precision in the numerical techniques with complicatedness that also raise quickly with the number of testing nodes [35]. To demonstrate our claims which partially motivated us to develop the present scheme for FPDEs, we compare it with a nonstandard finite difference (NSFD) method and standard finite difference (SFD) algorithm [27].To summarize, solving FPDEs with the proposed scheme offers the solution without coordinate transformations and computational cost does not grow immediately when the quantity of sampling points raises. Moreover, the product solutions constructed by the proposed methodology involve single independent variables regardless of the dimension of the problem and are continuous over all the domain of integration. Our scope in this paper is to give a promising and applicable algorithm for solving FPDEs to achieve an accurate solution which takes conveniences of the characteristics of the separation-variables scheme.The organization of this letter is as follows. To have the required mathematical background, in Section2 we describe some necessary definitions and mathematical preliminaries of the fractional calculus theory. In Section 3, we explain how the approximate-analytical method admitted by nonhomogeneous PDEs with fractional derivatives can be implemented. The usefulness of the above method has been illustrated through a number of examples in Section 4. In Section 5, we give a brief outline of our outcomes.


## 2. Notes on Fractional Calculus

Before embarking into the details of our method to FPDEs, we might want to review some fundamental definitions, results, and characteristics of the fractional calculus operators, utilized as a part of the remaining context of the report. The interested readers are refereed for more details to the following monographs [1, 2, 5, 17].Definition 1.
The Riemann-Liouville fractional integral operatorIα of order α≥0 is defined by(3)Itαux,t=1Γα∫0tt-τα-1ux,τdτ,α>0,I0ut=ut,in which Γ(·) indicates the gamma function.Definition 2.
Let us assumen is the smallest integer that is greater than α; the Caputo time-fractional derivative operator of order α>0 is defined as follows:(4)cDtαux,t=∂αux,t∂tα=1Γn-α∫0tt-τn-α-1∂nu,τ∂τndτ,n-1<α<n,∂nx,t∂tnα=n∈N.The Caputo-type fractional derivative is also stated ascDtαu(x,t)=Itn-α∂n(x,t)/∂tn. Furthermore, Caputo’s fractional derivatives are the linear operators given as follows:(5)cDtαηux,t+λvx,t=ηcDtαux,t+λcDtαvx,t,in which η and λ are constants.The Laplace transform of Caputo fractional derivative of orderα>0 is (as presented in [26, 27])(6)LDctαft=sαfs-∑k=0n-1sα-k-1fk0,n-1≤α≤n,n∈N.It is worthy to mention that(7)LDctαft=sαfs-sα-1f0,0<α≤1,in which(8)Lft=fs=∫0∞e-stftdt.


## 3. Formulation of the Solution Method

In this section we illustrate the strategy that is proposed to approximate the solution of the FPDEs with Caputo-type derivative. In this regard, the FPDE is converted to a PDE by approximating the fractional derivative using the approach proposed in [27]; then, a variable-separation technique is developed to achieve an approximate-analytical solution for the nonhomogeneous FPDE. In fact, the model reduction fulfilled at the first step can enhance the computational productivity.Let us consider the following Caputo-type FPDE as follows:(9)cDtαux,t=uxxx,t+gx,t,0<x<L,t>0,0<α<1,(10)ux,0=fx,0<x<L,u0,t=θt,uL,t=ωt,t>0.The Caputo fractional termcDtαu(x,t) is approximated by employing Laplace transform method and a linearization method proposed by Ren et al. [22] which was developed for interval arithmetic in [33] as follows:(11)LDctαux,t=sαux,s-sα-1ux,0,in which u(x,s) is the Laplace transform of u(x,t). In this study we assume that 0<α<1, so the term sα is linearized as(12)sα≈αs1+1-αs0=αs+1-α.By replacing (12) into (11) and using the inverse Laplace transform we have(13)cDtαux,t≈αut+1-αux,t-ux,0.By taking into account (13), the original FPDE (9) is simplified to the following PDE:(14)αutx,t+1-αux,t=uxxx,t+1-αfx+gx,t,0<x<L,t>0,0<α<1,(15)ux,0=fx,0<x<L,u0,t=θt,uL,t=ωt,t>0.In this step the problem is converted to a PDE and the fractional derivative is removed from the problem which can extremely reduce the computation cost. Now, we present our scheme to get the analytical solution of the PDE (14) which leads to getting an approximate-analytical solution of the FPDE (9).


### 3.1. Separation of Variables Method for Some PDEs

Let us consider the following PDE:(16)utx,t=c2uxxx,t,0<x<L,t>0,u0,t=uL,t=0,ux,0=fx.In order to find nontrivial solutions u(x,t) for the PDE (16), we should assume that u(x,t) is a product of two functions depending on two parameters t and x such that u(x,t)=F(x)G(t). Hence, we have ut=F(x)G˙(t) and uxx=F″(x)G(t). By substituting these products in (16) we have(17)FG˙=c2F″Gthat is implied as follows:(18)F″F=G˙c2G.Since the left-hand side of (18) is a function of x and the right side is a function of t, so it can be concluded that (18) holds such that(19)F″F=G˙c2G=k,in which k is constant. Therefore, by considering the boundary condition (16) we have(20)0=u0,t=F0Gt⟶F0=0,0=uL,t=FLGt⟶FL=0.Note that if we assume G(t)=0 in the above equations, it leads to a trivial solution u(x,t)=0. So, we have the next relations for F(x):(21)F″-kF=0,F0=FL=0.Now we determine the nontrivial solutions for different values of k. It is easy to find that (21) has a trial solution for k=0 and k>0. We have only nontrivial solution for k=-λ2<0 such that λ=nπ/L for all n∈N; thus Fn(x)=bnsin⁡(λx) is a solution for (21). By substituting k=-λ2=-(nπ/L)2 in G˙-kc2G=0, we obtain the solution Gn(t)=anexp(-λ2c2t) for all n∈N. Therefore, un(x,t)=Fn(x)Gn(t),∀n∈N is a solution of PDE (16) and the general solution of the problem can be formed as(22)ux,t=∑n=1AnGntFnx=∑n=1Anexp-ncπL2tsinnπLx.By applying the initial condition u(x,0)=f(x), we can specify the coefficients An as follows:(23)An=2L∫0LfxsinnπLxdx.Now let us consider the following problem:(24)utx,t+α2ux,t=β2uxxx,t+gx,t,0<x<L,t>0,u0,t=uL,t=0,ux,0=fx.In order to solve problem (24), at first we should consider the corresponding homogeneous equation ut(x,t)+α2u(x,t)=β2uxx(x,t). Similar to the procedure for problem (16), we assume the solution as u(x,t)=F(x)G(t). Therefore, by substituting u(x,t) into (16) we have(25)FG˙+α2FG=β2F″G⟶G˙+α2Gβ2G=F″F=k,where k is a constant. By taking into consideration the boundary condition F(0)=F(L)=0, it is proved that k=-λ2<0. Hence by solving the equation corresponding to F(x), it is implied that λ=nπ/L for all n∈N and we acquire Fn(x) as Fn(x)=Bnsin(nπ/Lx). Now we can get the solution of the nonhomogeneous problem (24) in a form as(26)ux,t=∑n=1GntsinnπLx.By replacing (26) into problem (24) we have(27)∑n=1G˙n+α2+βnπL2GnsinnπLx=gx,t.Consequently, the coefficients (G˙n+[α2+(βnπ/L)2]Gn) are the coefficients of Fourier sine series of g(x,t) given by(28)G˙n+α2+βnπL2Gn=2L∫0Lgx,tsinnπLxdx.In order to solve the above equation, we employ the initial condition u(x,0)=f(x) as follows:(29)fx=ux,0=∑n=1Gn0sinnπLx⟶Gn0=2L∫0Lfx,tsin⁡nπLxdx.Hence, by solving (28) based on the initial condition (29), we can obtain Gn(t) and finally u(x,t) is achieved, that is, the solution of problem (24).


## 3.1. Separation of Variables Method for Some PDEs

Let us consider the following PDE:(16)utx,t=c2uxxx,t,0<x<L,t>0,u0,t=uL,t=0,ux,0=fx.In order to find nontrivial solutions u(x,t) for the PDE (16), we should assume that u(x,t) is a product of two functions depending on two parameters t and x such that u(x,t)=F(x)G(t). Hence, we have ut=F(x)G˙(t) and uxx=F″(x)G(t). By substituting these products in (16) we have(17)FG˙=c2F″Gthat is implied as follows:(18)F″F=G˙c2G.Since the left-hand side of (18) is a function of x and the right side is a function of t, so it can be concluded that (18) holds such that(19)F″F=G˙c2G=k,in which k is constant. Therefore, by considering the boundary condition (16) we have(20)0=u0,t=F0Gt⟶F0=0,0=uL,t=FLGt⟶FL=0.Note that if we assume G(t)=0 in the above equations, it leads to a trivial solution u(x,t)=0. So, we have the next relations for F(x):(21)F″-kF=0,F0=FL=0.Now we determine the nontrivial solutions for different values of k. It is easy to find that (21) has a trial solution for k=0 and k>0. We have only nontrivial solution for k=-λ2<0 such that λ=nπ/L for all n∈N; thus Fn(x)=bnsin⁡(λx) is a solution for (21). By substituting k=-λ2=-(nπ/L)2 in G˙-kc2G=0, we obtain the solution Gn(t)=anexp(-λ2c2t) for all n∈N. Therefore, un(x,t)=Fn(x)Gn(t),∀n∈N is a solution of PDE (16) and the general solution of the problem can be formed as(22)ux,t=∑n=1AnGntFnx=∑n=1Anexp-ncπL2tsinnπLx.By applying the initial condition u(x,0)=f(x), we can specify the coefficients An as follows:(23)An=2L∫0LfxsinnπLxdx.Now let us consider the following problem:(24)utx,t+α2ux,t=β2uxxx,t+gx,t,0<x<L,t>0,u0,t=uL,t=0,ux,0=fx.In order to solve problem (24), at first we should consider the corresponding homogeneous equation ut(x,t)+α2u(x,t)=β2uxx(x,t). Similar to the procedure for problem (16), we assume the solution as u(x,t)=F(x)G(t). Therefore, by substituting u(x,t) into (16) we have(25)FG˙+α2FG=β2F″G⟶G˙+α2Gβ2G=F″F=k,where k is a constant. By taking into consideration the boundary condition F(0)=F(L)=0, it is proved that k=-λ2<0. Hence by solving the equation corresponding to F(x), it is implied that λ=nπ/L for all n∈N and we acquire Fn(x) as Fn(x)=Bnsin(nπ/Lx). Now we can get the solution of the nonhomogeneous problem (24) in a form as(26)ux,t=∑n=1GntsinnπLx.By replacing (26) into problem (24) we have(27)∑n=1G˙n+α2+βnπL2GnsinnπLx=gx,t.Consequently, the coefficients (G˙n+[α2+(βnπ/L)2]Gn) are the coefficients of Fourier sine series of g(x,t) given by(28)G˙n+α2+βnπL2Gn=2L∫0Lgx,tsinnπLxdx.In order to solve the above equation, we employ the initial condition u(x,0)=f(x) as follows:(29)fx=ux,0=∑n=1Gn0sinnπLx⟶Gn0=2L∫0Lfx,tsin⁡nπLxdx.Hence, by solving (28) based on the initial condition (29), we can obtain Gn(t) and finally u(x,t) is achieved, that is, the solution of problem (24).


## 4. Numerical Experiments

To reveal the usefulness of the proposed plan for the nonhomogeneous PDEs with the Caputo-type derivative, we test a number of FPDEs and compare with the NSFD method and SFD technique [27]. The accuracy of these methods is computed by a maximum norm relative error [27] and error norm L∞. It is shown that the present technique is a precise and cost efficient tool for the solution of FPDEs. The numerical computations are implemented through Matlab software, version R2010a, and the CPU of system is Intel(R) core(TM) i3-4130 with RAM 4 GB.Remark 3.
To have an accurate comparison with the technique presented in [27], we employ the nonstandard Crank-Nicolson method as a NSFD method and standard Crank-Nicolson method as a SFD technique with the formula proposed in [27].Example 4.
Let us consider the following FPDE [27]:(30)cDtαux,t=uxxx,t+6Γ4-αt3-α+t3+1sin⁡x,ux,0=sin⁡x,0<x<π,0<α<1,u0,t=uπ,t=0.The exact solution of the above problem isu(x,t)=(t3+1)sin⁡x. By employing the technique introduced in [27], we approximate the Caputo fractional derivative; hence, the problem (30) is converted to the following system:(31)αutx,t+1-αux,t=uxxx,t+6Γ4-αt3-α+t3+2-αsin⁡x,0≤x≤π,t≥0,ux,0=sinx,u0,t=uπ,t=0.Since the boundary conditions are homogeneous, we can obtain the solution of (31) by using the separation of variables method. So, let us assume that(32)ux,t=∑n=1Gntsin⁡nx,by replacing (32) in (31) we have(33)∑n=1αG˙nt+1-α+n2Gntsinnx=6Γ4-αt3-α+t3+2-αsinx,in which G˙n(t) is the derivative of Gn(t) with respect to t.Regarding the equivalency of the right-hand side of (33) with the left-hand side, it can be implied that αG˙n(t)+(1-α+n2)Gn(t) are the coefficients of the Fourier sin series of the right-hand side function. So, we have for n=1(34)αG˙1t+2-αG1t=6Γ4-αt3-α+t3+2-αand for n≥2(35)αG˙nt+1-α+n2Gnt=0.From the initial condition of problem (31) we have(36)sinx=ux,0=∑n=1Gn0sinnx⟹G10=1,∀n≥0,Gn0=0.By solving the two systems of equations,(37)αG˙1t+2-αG1t=6Γ4-αt3-α+t3+2-α,G10=1,αG˙nt+1-α+n2Gnt=0,Gn0=0,we can find that G1(t)≠0, Gn(t)=0 for n≥2 and(38)ux,t=G1tsin⁡x.In [27], the authors employed the following SFD method to approximate the solution of problem (31):(39)αuin+1-uinΔt+1-α2uin+uin+1=12ui+1n-2uin+ui-1nΔ2x+ui+1n+1-2uin+1+ui-1n+1Δ2x+Rin+Rin+12,in which(40)Rx,t=6Γ4-αt3-α+t3+2-αsin⁡x.Now we solve problem (31) by using the NSFD method. Note that the formula of NSFD method is similar to the SFD method (39) with a difference that Δt is replaced with ϕ(Δt)=exp(βΔt)-1/β, where β=α-2/α. Hence, an algebraic nonlinear equations system is obtained that can be solved by LU decomposition method to get the approximate solution. Then, the relative error in maximum norm between the numerical solution and the exact solution of (31) is achieved using the following formula:(41)EΔx,Δt=max0≤n≤N⁡max1≤j≤M-1⁡uxj,tn-ujnmax1≤j≤M-1⁡uxj,tn.In Table1 by using formula (41), the difference between the approximate solutions of the NSFD method, SFD method, and the proposed approximate-analytical solution is compared with the exact solution of the original FPDE (30) for Δx=π/100,Δt=1/100 on t∈[0,1]. A comparison of the NSFD and SFD solutions based on the error norm L∞ for different values of α at T=1 is shown in Table 2. The numerical results illustrated that the present scheme works very well for this problem, specially near α=1. The order of accuracy is approximately the same for all the numerical methods. However, there is no requirement to check the stability and consistency of the solution for the proposed algorithm while for others it is critical. Moreover, the graphs of the numerical solution (38) with the error function between the exact and numerical solutions over t∈[0,1] for α=0.99 are plotted in Figure 1.Table 1
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example4.
α
Our scheme
NSFD method
SFD method [27]
0.01
0.002142897
0.001377269
0.002101727
0.1
0.019659612
0.019617338
0.019617785
0.3
0.045194043
0.045149718
0.045148926
0.5
0.051662902
0.051612374
0.051614489
0.7
0.041504057
0.041454967
0.041453574
0.9
0.017072953
0.017021427
0.017021427
0.99
0.001846904
0.001795412
0.001795165Table 2
Error normL∞ for Example 4.
α
Our scheme
NSFD method
SFD method [27]
0.01
0.003699
0.001490
0.003617
0.1
0.034897
0.033022
0.034801
0.3
0.084289
0.082783
0.084514
0.5
0.098251
0.098184
0.099550
0.7
0.078375
0.080241
0.081020
0.9
0.029823
0.033207
0.033430
0.99
0.000609
0.003509
0.003529Figure 1
The numerical solution (a) with the error function (b) of Example4 for α=0.99 over t∈0,1.
(a)
(b)Example 5.
Let us consider the following FPDE [27]:(42)cDtαux,t=uxxx,t+16Γ4+αt3-t3+αsin⁡x,ux,0=1,0<x<π,u0,t=uπ,t=1.The the exact solution isu(x,t)=t3+αsin⁡x+1. Similar to problem (30), the above problem is converted to the following PDE:(43)αutx,t+1-αux,t=uxxx,t+1-α+Γ4+α6t3-t3+αsinx,0<x<π,ux,0=1,u0,t=uπ,t=1.To change the conditions of problem (43) to homogeneous conditions, we replace u(x,t) with v(x,t)+1. By substituting this new variable in (43) we have(44)αvtx,t+1-αvx,t=vxxx,t+Γ4+α6t3-t3+αsinx,0<x<π,vx,0=0,v0,t=vπ,t=0.Similar to Example 4, we can get the general solution of problem (44) according the boundary conditions as(45)vx,t=∑n=1Gntsin⁡nx.By replacing (45) into (44) we have(46)∑n=1αG˙nt+1-α+n2Gntsin⁡nx=Γ4+α6t3-t3+αsin⁡x.Hence, αG˙n(t)+(1-α+n2)Gn(t) are the coefficients of the Fourier sin series of the right-hand side function and we have for n=1(47)αG˙1t+2-αG1t=Γ4+α6t3-t3+αand for n≥2(48)αG˙nt+1-α+n2Gnt=0.Regarding the initial conditions (43), we have(49)0=vx,0=∑n=1Gn0sinnx⟹Gn0=0∀n.By solving the systems of equations,(50)αG˙1t+2-αG1t=Γ4+α6t3-t3+α,G10=0,αG˙nt+1-α+n2Gnt=0,Gn0=0,it is obtained that G1(t)≠0, Gn(t)=0 for n≥2 and(51)vx,t=G1tsint,ux,t=G1tsint+1.The exact solution of the original FPDE is tα+3sin(x)+1. As it is obvious, the difference between the exact solution and the numerical solution u(x,t) is the terms G1(t) and t3+α. Jiang and Ma [27] approximated the solution using the SFD method (39) such that(52)Rx,t=1-α+Γ4+α6t3-t3+αsinx,and the error of the method is analyzed by using relation (41). Once again we employ NSFD method to have a comparison among our proposed scheme, NSFD method, and SFD technique. Indeed, the formula of NSFD method is similar to the SFD method (39) with a difference that Δt is replaced with ϕ(Δt)=exp(βΔt)-1/β in which β=α-2/α. We also employ (41) to compare the numerical solution u(x,t) with the the SFD method and NSFD techniques by assuming Δx=π/100,Δt=1/100 on t∈[0,1] in Table 3. In addition, error norm L∞ is used to have a comparison from another point of view that is illustrated in Table 4. From the tables we can see that the order of accuracy for all the present numerical techniques is the same. Besides, error norm L∞ for the approximate-analytical solution is plotted with α=0.9,0.99 and 0.999 at T=1 in Figure 2.Table 3
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example5.
α
Our scheme
NSFD method
SFD method [27]
0.01
0.4968555
0.0741259
0.4968554
0.1
0.4683680
0.3992456
0.4683675
0.3
0.4063310
0.3861797
0.4063114
0.5
0.3461901
0.3362421
0.3461852
0.7
0.2833040
0.2776980
0.2832938
0.9
0.2117990
0.2085503
0.2117820
0.99
0.1741744
0.1726555
0.1751539Table 4
Error normL∞ for Example 5.
α
Our scheme
NSFD method
SFD method [27]
0.01
0.993711
0.080130
0.993710
0.1
0.936737
0.698034
0.936735
0.3
0.812629
0.738043
0.812623
0.5
0.692474
0.655328
0.692337
0.7
0.566690
0.546031
0.566587
0.9
0.423630
0.412194
0.423564
0.99
0.350385
0.341835
0.350307Figure 2
Error normL∞ for Example 5 (α=0.9-□; α=0.99-∘, and α=0.999-×) at T=1.Remark 6.
It is worth noting here that the results of this problem, by using the SFD method as it is shown in [27], are not correct. Thereby, we modified the results to have a fair comparison for this example.Example 7.
Consider the following FPDE [27]:(53)cDtαux,t=uxxx,t+6Γ4-αt3-α-t3-1ex,ux,0=ex,0<x<1,u0,t=t3+1,u1,t=t3+1e,t>0.The analytical solution of problem (53) is u(x,t)=(t3+1)ex and 0<α<1. Once again the Caputo-type derivative is approximated by the described technique in the previous section. Therefore, the problem changes to the following PDE:(54)αutx,t+1-αux,t=uxxx,t+1-αex+6Γ4-αt3-α-t3-1ex,0<x<1,ux,0=ex,u0,t=t3+1,u1,t=1+t3e.In order to change Example (54) to a problem with the homogeneous conditions, we use a change of variables as u(x,t)=v(x,t)+(t3+1)ex. Therefore, we have(55)αvtx,t+1-αvx,t=vxxx,t+6Γ4-αt3-α-1-αt3-3αt2ex,0<x<1,vx,0=0,v0,t=0,u1,t=0.Analogously to the demonstration of the procedure for the previous examples, we can consider the solution of (55) as v(x,t)=∑n=1‍Gn(t)sin(nπx). By substituting it into (55) we have(56)∑n=1αG˙nt+1-α+n2π2Gntsinnπx=6Γ4-αt3-α-1-αt3-3αt2ex.Therefore, we have(57)αG˙nt+1-α+n2π2Gnt=2∫016Γ4-αt3-α-1-αt3-3αt2exsinnπxdx=26Γ4-αt3-α-1-αt3-3αt2nπ1--1nen2π2+1.Now by solving the above differential equation we can getGn(t) and finally v(x,t) as(58)vx,t=∑n=1Gntsinnπx,ux,t=∑n=1Gntsinnπx+t3+1ex.From the exact solution of the original FPDE (53), it can be seen that the only difference between the approximate solution, (58), and exact solutions is v(x,t)=∑n=1‍Gn(t)sin(nπx). Table 5 depicts the maximum norm relative error based on formula (41) for the approximate-analytical solution (58), NSFD method, and SFD technique at Δx=1/100,Δt=1/100 at T=1. Similar to the previous examples, the authors in [27] solved problem (54) by using the SFD method (39) such that(59)Rx,t=1-αex+6Γ4-αt3-α-t3-1ex.Note that the NSFD method is similar to the SFD formula (39) with a difference that Δx is substituted by sin(πΔx)/π and Δt is replaced with ϕ(Δt)=exp(βΔt)-1/β in which β=α-1-π2/α.Table 5
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example7.
α
Our scheme
NSFD method
SFD method [27]
0.01
3.15
e
-
4
2.38
e
-
5
3.02
e
-
4
0.1
2.84
e
-
3
2.60
e
-
3
2.82
e
-
3
0.3
6.97
e
-
3
6.89
e
-
3
6.91
e
-
3
0.5
8.72
e
-
3
8.62
e
-
3
8.65
e
-
3
0.7
7.69
e
-
3
7.60
e
-
3
7.62
e
-
3
0.9
3.45
e
-
3
3.40
e
-
3
3.42
e
-
3
0.99
3.88
e
-
4
3.57
e
-
4
3.81
e
-
4In this experiment, maximum norm relative errors of the numerical methods for different choices ofα on t∈[0,1] are displayed in Table 5. Also, the comparison of the numerical results by using the error norm L∞ at T=1 for different values of α is illustrated in Table 6. We recall that the proposed technique has good accuracy and robustness for different values of α in comparison with others. Also, the error norm L∞ of the proposed approximate-analytical solution is also plotted for α=0.8, 0.9, and 0.99 at T=1 in Figure 3.Table 6
Error normL∞ for Example 7.
α
Our scheme
NSFD method
SFD method [27]
0.01
1.416
e
-
3
2.385
e
-
5
1.413
e
-
3
0.1
1.310
e
-
2
8.690
e
-
3
1.309
e
-
2
0.3
3.169
e
-
2
2.919
e
-
2
3.168
e
-
2
0.5
3.897
e
-
2
3.770
e
-
2
3.896
e
-
2
0.7
3.367
e
-
2
3.316
e
-
2
3.365
e
-
2
0.9
1.475
e
-
2
1.460
e
-
2
1.474
e
-
2
0.99
1.630
e
-
3
1.512
e
-
3
1.620
e
-
3Figure 3
Error normL∞ for Example 7 (α=0.8-□; α=0.9-∘, and α=0.99-+) at T=1.Example 8.
We consider the time-fractional PDE as follows [27]:(60)cDtαux,t=uxxx,t+6Γ4-αt3-α+t3cos⁡x-ex,ux,0=ex,0<x<1,u0,t=t3+1,u1,t=t3cos⁡1+e,and the exact solution of the above problem is u(x,t)=t3cos(x)+ex.In a similar fashion, we approximate the fractional derivative to covert the time-fractional PDE (60) to the following PDE:(61)αutx,t+1-αux,t=uxxx,t+6Γ4-αt3-α+t3cos⁡x-αex,0<x<1,t>0,ux,0=ex,u0,t=t3+1,u1,t=t3cos⁡1+e.We define the change of variableu(x,t)=v(x,t)+(t3cos⁡x+ex) and replace it in the above system. Hence, we need to solve the resulting system as follows:(62)αvtx,t+1-αvx,t=vxxx,t+6Γ4-αt3-α-3αt2+α-1t2cos⁡x,0<x<1,t>0,vx,0=0,v0,t=v1,t=0.According to the above system, we can assume the solution of the problem in the form as v(x,t)=∑n=1‍Gn(t)sin(nπx). By substituting this product solution in system (62), we have(63)∑n=1αG˙nt+1-α+n2π2Gntsinnπx=6Γ4-αt3-α-3αt2+α-1t2cos⁡x.Therefore, we have(64)αG˙nt+1-α+n2π2Gnt=2∫016Γ4-αt3-α-3αt2+α-1t2cosxsinnπxdx=6Γ4-αt3-α-3αt2+α-1t22nπn2π2-11--1ncos⁡1.Regarding the condition 0=v(x,0)=∑n=1‍Gn(t)sin(nπx), we have Gn(0)=0. To obtain Gn for n≥1, we need to solve the following system:(65)αG˙nt+1-α+n2π2Gnt=6Γ4-αt3-α-3αt2+α-1t22nπn2π2-11--1ncos⁡1,Gn0=0.Note that the NSFD method here is the same as the formula for Example7. The accuracy of the proposed method is tested by using the relative error formula (41) over t∈[0,1] and error norm L∞ for different values of α at T=1 in Tables 7 and 8, respectively. Undoubtedly these results also exhibit the accuracy and proficiency of the present method for this problem. Also, the error norm L∞ was obtained between the approximate-analytical solution and the exact solution for various values of for α at T=1 in Figure 4. We see that we can accomplish a suitable approximation with the analytical solution by applying the proposed method, taking into account the computational costs of the discretization of domain into finite points in the existing numerical schemes.Table 7
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example8.
α
Our scheme
NSFD method
SFD method [27]
0.01
2.29
e
-
4
2.38
e
-
5
2.21
e
-
4
0.1
2.13
e
-
3
1.54
e
-
3
2.05
e
-
3
0.3
5.17
e
-
3
4.87
e
-
3
4.98
e
-
3
0.5
6.40
e
-
3
6.15
e
-
3
6.17
e
-
3
0.7
5.57
e
-
3
5.35
e
-
3
5.37
e
-
3
0.9
2.47
e
-
3
2.36
e
-
3
2.37
e
-
3
0.99
2.66
e
-
4
2.46
e
-
4
2.63
e
-
4Table 8
Error normL∞ for Example 8.
α
Our scheme
NSFD method
SFD method [27]
0.01
7.19
e
-
4
4.45
e
-
5
7.17
e
-
4
0.1
6.65
e
-
3
4.39
e
-
3
6.65
e
-
3
0.3
1.61
e
-
2
1.48
e
-
2
1.60
e
-
2
0.5
1.98
e
-
2
1.91
e
-
2
1.97
e
-
2
0.7
1.71
e
-
2
1.68
e
-
2
1.70
e
-
2
0.9
7.49
e
-
3
7.43
e
-
3
7.49
e
-
3
0.99
8.31
e
-
4
7.79
e
-
4
8.23
e
-
4Figure 4
Error normL∞ for Example 8 (α=0.8-□; α=0.9-∘; and α=0.95-×) at T=1.Example 9.
Let us consider the FPDE as follows:(66)cDt1/2ux,t=uxxx,t+x1-x6t2.5Γ3.5+2t3+1,t>0ux,0=xx-1,0<x<1,u0,t=u1,t=0,and the exact solution of the above problem is u(x,t)=x(x-1)(t3+1).Once again we approximate the fractional derivative using the plan explained in Section3. Therefore, the FPDE (66) is converted to the following PDE:(67)utx,t+ux,t=2uxxx,t+x1-x+2x1-x6t2.5Γ3.5+4t3+1,0<x<1,t>0,ux,0=xx-1,u0,t=u1,t=0.Since the boundary conditions are homogeneous, we can hypothesize the solution as u(x,t)=∑n=1‍Gn(t)sin(nπx) and replace it in the above system. So, we have(68)∑n=1G˙nt+1+2n2π2Gntsinnπx=x1-x+2x1-x6t2.5Γ3.5+4t3+1.Hence, we have(69)G˙nt+1+2n2π2Gnt=2∫01x1-x+2x1-x6t2.5Γ3.5+4t3+1sinnπxdx=41--1nn3π31+12t2.5Γ3.5+81--1nnπt3+1.By considering the assumption x(1-x)=u(x,0)=∑n=1‍Gn(0)sin(nπx), we have(70)Gn0=2∫01x1-xsinnπxdx=41--1nn3π3,and to find Gn(t), the following system should be solved:(71)G˙nt+1+2n2π2Gnt=41--1nn3π31+12t2.5Γ3.5+81--1nnπt3+1,Gn0=41--1nn3π3.It is easy to see that Gn(t)=0 for n is even.It is worth noting that the demonstration of NSFD method is analogous to the SFD formula (39) with a difference that Δt is replaced with ϕ(Δt)=exp⁡(βΔt)-1/β in which β=(-1/2-n2)/1/2.In Table9, we compute the maximum norm relative errors and error norm L∞ for approximate-analytical scheme, NSFD method, and SFD technique. The numerical results demonstrate the appropriateness of the present technique for this case and good accuracy in comparison with the others. Besides that, to observe the behavior of the approximate solution over the time interval, we plot the absolute error of the method for t=0.25,0.5,0.75, and 1 in Figure 5. From the observation, the method has a smooth behavior over the time domain and obtains a satisfactory result.Table 9
Maximum norm relative errors and error normL∞ of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 9.
Our scheme
NSFD method
SFD method [27]
Relative error
0.01178
0.01158
0.01158
L
∞
0.004797
0.004656
0.004795Figure 5
Absolute errors for different values oft, Example 9.


## 5. Concluding Remarks

This article deals with the numerical solution of nonhomogeneous time-fractional PDEs by an approximate-analytical method based on a separation-variables technique. A comparative study is presented by applying two different finite difference methods and the proposed techniques. The strategy exploits an approximating process to reduce the FPDE to the corresponding PDE. This combination is rational and efficient, because, on one hand, the separation-variables technique is very suitable to the homogeneous PDE and, more importantly, can provide an exact solution and reduce the computational cost by avoiding extra conditions on the stability conditions of the method, and on the other hand the adaptive method can cope with different kinds of FPDEs with less effort. The utilization of approximating technique for the fractional derivative radically decreases the computational expense of this variable-separation algorithm. The results of numerical examples demonstrate that this method is accurate similar to finite difference methods.



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