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

1Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran 2Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran 3Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 4Department of Mathematics, Çankaya University, Balgat, 06530 Ankara, Turkey 5Institute of Space Sciences, Magurele, Bucharest, Romania


Introduction
Many anomalous diffusion processes which existed in some physical and biological areas can be modeled by the timefractional 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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][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 2 Complexity 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 order  ∈ (0, 1] as      (, ) −   (, ) =  () ,  ∈ [0, ] ,  > 0 (1) subject to the initial and boundary conditions: (, 0) =  () , 0 <  < ,  (0, ) =  () ,  (, ) =  () ,  > 0. ( The analytical solutions of the FPDEs were investigated in literature by employing of Green's functions or Fourier-Laplace transforms [24][25][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][29][30][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 steppingtype 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][33][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 Section 2 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.

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 operator I  of order  ≥ 0 is defined by in which Γ(⋅) indicates the gamma function.
Definition 2. Let us assume  is the smallest integer that is greater than ; the Caputo time-fractional derivative operator of order  > 0 is defined as follows: The Caputo-type fractional derivative is also stated as  D   (, ) = I −  (  (, )/  ).Furthermore, Caputo's fractional derivatives are the linear operators given as follows: D   ( (, ) + V (, )) in which  and  are constants.

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: D    (, ) =   (, ) +  (, ) , 0 <  < ,  > 0, 0 <  < 1, (, 0) =  () , 0 <  < ,  (0, ) =  () ,  (, ) =  () ,  > 0. ( The Caputo fractional term  D   (, ) 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: in which (, ) is the Laplace transform of (, ).In this study we assume that 0 <  < 1, so the term   is linearized as By replacing (12) into (11) and using the inverse Laplace transform we have By taking into account (13), the original FPDE ( 9) is simplified to the following PDE: 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).

Separation of Variables Method for Some PDEs. Let us consider the following PDE:
In order to find nontrivial solutions (, ) for the PDE (16), we should assume that (, ) is a product of two functions depending on two parameters  and  such that (, ) = F()G().Hence, we have   = F() Ġ() and   = F  ()G().By substituting these products in (16) we have that is implied as follows: Since the left-hand side of ( 18) is a function of  and the right side is a function of , so it can be concluded that ( 18) holds such that in which  is constant.Therefore, by considering the boundary condition (16) we have

Complexity
Note that if we assume G() = 0 in the above equations, it leads to a trivial solution (, ) = 0. So, we have the next relations for F(): Now we determine the nontrivial solutions for different values of .It is easy to find that ( 21) has a trial solution for  = 0 and  > 0. We have only nontrivial solution for  = − 2 < 0 such that  = / for all  ∈ N; thus F  () =   sin() is a solution for (21).By substituting 16) and the general solution of the problem can be formed as By applying the initial condition (, 0) = (), we can specify the coefficients   as follows: Now let us consider the following problem: In order to solve problem (24), at first we should consider the corresponding homogeneous equation   (, ) +  2 (, ) =  2   (, ).Similar to the procedure for problem (16), we assume the solution as (, ) = F()G().Therefore, by substituting (, ) into ( 16) we have where  is a constant.By taking into consideration the boundary condition F(0) = F() = 0, it is proved that  = − 2 < 0. Hence by solving the equation corresponding to F(), it is implied that  = / for all  ∈ N and we acquire F  () as F  () =   sin((/)).Now we can get the solution of the nonhomogeneous problem (24) in a form as By replacing (26) into problem (24) we have Consequently, the coefficients ( Ġ + [ 2 + (/) 2 ]G  ) are the coefficients of Fourier sine series of (, ) given by In order to solve the above equation, we employ the initial condition (, 0) = () as follows: Hence, by solving (28) based on the initial condition (29), we can obtain G  () and finally (, ) is achieved, that is, the solution of problem (24).

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  ∞ .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]: The exact solution of the above problem is (, ) = ( 3 + 1) sin .By employing the technique introduced in [27], we approximate the Caputo fractional derivative; hence, the problem ( 30) is converted to the following system: 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 by replacing (32) in (31) we have in which Ġ () is the derivative of G  () with respect to .
Regarding the equivalency of the right-hand side of (33) with the left-hand side, it can be implied that  Ġ () + (1 − + 2 )G  () are the coefficients of the Fourier sin series of the right-hand side function.So, we have for  = 1 and for  ≥ 2 From the initial condition of problem (31) we have By solving the two systems of equations, we can find that G 1 () ̸ = 0, G  () = 0 for  ≥ 2 and In [27], the authors employed the following SFD method to approximate the solution of problem (31): in which 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 Δ is replaced with (Δ) = (exp(Δ) − 1)/, where  = ( − 2)/.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: In Table 1 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 Δ = /100, Δ = 1/100 on  ∈ [0, 1].A comparison of the NSFD and SFD solutions based on the error norm  ∞ for different values of  at  = 1 is shown in Table 2.The numerical results illustrated that the present scheme works very well for this problem, specially near  = 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  ∈ [0, 1] for  = 0.99 are plotted in Figure 1.
Example 5. Let us consider the following FPDE [27]: The the exact solution is (, ) =  3+ sin  + 1.Similar to problem (30), the above problem is converted to the following PDE:  Regarding the initial conditions (43), we have By solving the systems of equations, it is obtained that G 1 () ̸ = 0, G  () = 0 for  ≥ 2 and The exact solution of the original FPDE is  +3 sin() + 1.As it is obvious, the difference between the exact solution and the numerical solution (, ) is the terms G 1 () and  3+ .Jiang and Ma [27] approximated the solution using the SFD method (39) such that 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 Δ is replaced with (Δ) = (exp(Δ) − 1)/ in which  = ( − 2)/.We also employ (41) to compare the numerical solution (, ) with the the SFD method and NSFD techniques by assuming Δ = /100, Δ = 1/100 on  ∈ [0, 1] in Table 3.In addition, error norm  ∞ 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  ∞ for the approximate-analytical solution is plotted with  = 0.9, 0.99 and 0.999 at  = 1 in Figure 2.
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.( The analytical solution of problem (53) is (, ) = ( 3 + 1)  and 0 <  < 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: In order to change Example (54) to a problem with the homogeneous conditions, we use a change of variables as (, ) = V(, ) + ( 3 + 1)  .Therefore, we have Analogously to the demonstration of the procedure for the previous examples, we can consider the solution of (55) as V(, ) = ∑ =1 G  ()sin().By substituting it into (55) we have Therefore, we have Now by solving the above differential equation we can get G  () and finally V(, ) as 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(, ) = ∑ =1 G  ()sin().Table 5 depicts the maximum norm relative error based on formula (41) for the approximateanalytical solution (58), NSFD method, and SFD technique at Δ = 1/100, Δ = 1/100 at  = 1.Similar to the previous examples, the authors in [27] solved problem (54) by using the SFD method (39) such that Note that the NSFD method is similar to the SFD formula (39) with a difference that Δ is substituted by sin(Δ)/ and Δ is replaced with (Δ) = (exp(Δ) − 1)/ in which  = ( − 1 −  2 )/.In this experiment, maximum norm relative errors of the numerical methods for different choices of  on  ∈ [0, 1] are displayed in Table 5.Also, the comparison of the numerical results by using the error norm  ∞ at  = 1 for different values of  is illustrated in Table 6.We recall that the proposed technique has good accuracy and robustness for different values of  in comparison with others.Also, the error norm  ∞ of the proposed approximate-analytical solution is also plotted for  = 0.8, 0.9, and 0.99 at  = 1 in Figure 3.
and the exact solution of the above problem is (, ) =  3 cos() +   .
In a similar fashion, we approximate the fractional derivative to covert the time-fractional PDE (60) to the following PDE: We define the change of variable (, ) = V(, ) + ( 3 cos  +   ) and replace it in the above system.Hence, we need to solve the resulting system as follows: According to the above system, we can assume the solution of the problem in the form as V(, ) = ∑ =1 G  ()sin().
By substituting this product solution in system (62), we have Therefore, we have Regarding the condition 0 = V(, 0) = ∑ =1 G  ()sin(), we have G  (0) = 0. To obtain G  for  ≥ 1, we need to solve the following system: Note that the NSFD method here is the same as the formula for Example 7. The accuracy of the proposed method is tested by using the relative error formula (41) over  ∈ [0, 1] and error norm  ∞ for different values of  at  = 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  ∞ was obtained between the approximate-analytical solution and the exact solution for various values of for  at  = 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.
Once again we approximate the fractional derivative using the plan explained in Section 3. Therefore, the FPDE (66) is converted to the following PDE: Since the boundary conditions are homogeneous, we can hypothesize the solution as (, ) = ∑ =1 G  ()sin() and replace it in the above system.So, we have Hence, we have (71) It is easy to see that G  () = 0 for  is even.
It is worth noting that the demonstration of NSFD method is analogous to the SFD formula (39) with a difference that Δ is replaced with (Δ) = (exp(Δ) − 1)/ in which  = (−1/2 −  2 )/(1/2).In Table 9, we compute the maximum norm relative errors and error norm  ∞ 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  = 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.

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.

Figure 5 :
Figure 5: Absolute errors for different values of , Example 9.

Table 1 :
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 4.

Table 3 :
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 5.

Table 5 :
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 7.

Table 7 :
Maximum norm relative errors of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 8.

Table 9 :
Maximum norm relative errors and error norm  ∞ of the proposed approximate-analytical solution, NSFD method, and SFD technique for Example 9.