Toward computational algorithm for time-fractional Fokker–Planck models

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.


Introduction
Recent decades have witnessed great attention toward fractional calculus, which can be considered as a generalization of classical integer-order integration and differentiation. Many definitions have been suggested for fractional derivatives such as Riesz, Riemann-Liouville, Grunwald-Letnikov, Caputo, and conformable fractional definitions. [1][2][3][4] Such differential and integral operators of non-integer order include all historical states of the function in a weighted form called the memory effect. Anyhow, a large number of physical systems are modeled using fractional differential equations (FDEs), particularly fractional partial differential equations (FPDEs). The FPDEs have achieved significance and publicity due to their tremendous use in different fields such as electrochemistry, electrical circuits, theoretical biology, and quantum mechanics. [5][6][7][8] Furthermore, the most significant feature for using the FPDEs in such and other applications is the non-local property, while the differential operator of integer order is local. In this light, since the next state of a fractional system depends not only on the current state but also on its entire historical states. This leads to deep consistency of the mathematical model components in dynamic systems and physical processes. Nevertheless, solving those FDEs is a challenge, especially for numerical calculations. Thus, an effective, reliable, and appropriate numerical methods are needed in handling the partial differential equations (PDEs) with a fractional order of physical interest.
The Fokker-Planck equation is one of the classical widely used equations of statistical physics, which was first presented by Fokker and Planck for describing the Brownian motion for particles and the change of probability of a random function in space and time. 9 In addition, chemical Fokker-Planck equation can be derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation. This equation turns out to be more accurate than the linear-noise approximation of the chemical master equation. Anyhow, Fokker-Planck equation arises in the modeling of many natural science phenomena, including quantum optics, electron relaxation, polymer dynamics, solid-state system, probability flux, and other theoretical and practical models. 10 The concern of this analysis is to consider the numerical approximate solutions of the Fokker-Planck PDE with time-fractional derivative of the following form along with the initial condition where D b t is the Caputo time-fractional derivatives,v 0 (x) is the given analytic function on R, and v(x, t) is suggested to be a causal function of space and time, which means that v(x, t) is vanishing for x\0 and t\0. The smooth functions f 1 (x, t) and f 2 (x, t) are the diffusion and drift coefficients, respectively. Under such assumptions, we assume that the initial value problem (IVP) (1) and (2) has a unique analytical solution on the domain of interest.
The fractional Fokker-Planck equation (F-FPE) has been successfully used in biological molecules, chemical physics, energy consumption, and engineering. Indeed, fractional diffusion, a specific type of F-FPE, has been also applied to several situations such as frequencydependent damping behavior of materials, viscoelasticity, and diffusion processes. 9 Unfortunately, it is not easy to obtain the exact solution for FDEs in general. So, many numerical and analytical techniques are employed to approximate these solutions. The multistep reduced differential transform method, 10 the predictor-corrector approach, 11 the Laplace transform method, 12 the variational iteration method (VIM), 13 and Adomain decomposition method (ADM) 13 are some of the advanced numerical and approximate methods that have been applied for F-FPEs.
In this article, the residual power series (RPS) method is implemented for solving IVPs (1) and (2). The RPS technique was developed to approximate solutions for certain class of fuzzy differential equations. 14 Later, it was applied in solving different types of differential equations due to its simplicity, accuracy, and efficiency. [15][16][17][18][19] The residual power series method (RPSM) has many advantages; first, it is easy to construct a power series solution for handling both linear and nonlinear equations without the terms of linearization, discretization, or perturbation. Second, the present method provides the solutions in Taylor expansions; therefore, the exact solutions will be available when the solutions are polynomials. [20][21][22][23][24] This technique is a direct way to ensure the rate of convergence for series solution, as it depends on minimizing the residual error related. Third, the solutions along with their derivatives can be applied for each arbitrary point in the given interval. Fourth, the RPSM does not require modifications while converting from lower to higher order. Consequently, it has to be easily applied to the proposed system by selecting an appropriate value for the initial guesses approximations. Fifth, the RPS technique needs minor computational requirements with less time and more accuracy. In addition, the presented method is not affected by round-off errors, since it gives the solution in a form of fractional power series (FPS), and substituting values for the solution variables happens as a final step. Finally, it is of global nature in terms of the solutions obtained as well as its ability to solve various types of mathematical, physical, and engineering problems. [25][26][27][28][29] The rest of this work is organized as follows. In section ''Preliminaries and notations,'' some essential definitions and primary results relating to fractional calculus are given. In sections ''Description of the RPS algorithm,'' the RPS algorithm is presented for solving time-fractional Fokker-Planck equations (TF-FPEs). Numerical and analytical results for some illustrative examples using the RPS algorithm are introduced in section ''Numerical experiments.'' Meanwhile, numeric comparison between the proposed method and those available in the literature is discussed. Concluding remarks are given in the last section.

Preliminaries and notations
In this section, we revisit some essential definitions and basic properties of popular fractional operators, Riemann-Liouville fractional integral and Caputo fractional derivative. Then, we survey the most important results of the FPS representation. Throughout this analysis, the set of real numbers and the set of natural numbers are denoted by R and N, respectively, while the gamma function is denoted by G.
Definition 2.1. The integral operator for Riemann-Liouville of order b ø 0 is given by 2 where I is the domain of interest for x. Next, we present the Caputo fractional derivative 3 of order b, which is an alternative operator to the Riemann-Liouville fractional operator as follows In fact, Caputo fractional derivative allows us to include the classical initial and boundary conditions in the formulation of the model, whereas the derivative of a constant is 0. For such reasons, the Caputo sense is considered in this analysis to handle the Fokker-Planck equation.
Definition 2.2. For n À 1\b\n, n 2 N. The Caputo time-fractional derivative operator of order b is defined by 1 Similarly, the Caputo space-fractional derivative operator of order b is defined by Theorem 2.1. If n À 1\a ł n and n 2 N, then The following are some properties of the operators D b a and J b a for b.0,q. À 1, that can be found in Definition 2.
3. An FPS representation at t 0 has the following form 19 where 0 ł n À 1\b ł n, x 2 I and t ø t 0 is called multiple fractional power series (MFPS) about t 0 .
where r is a positive real number, called the radius of convergence for the MFPS.

Description of the RPS algorithm
The main goal of this section is to present the methodology of the RPS technique in obtaining the MFPS approximation of the time-fractional Fokker-Planck model based on the formula of generalized Taylor in Caputo sense by providing a fractional recursion formula to obtain the coefficients of the MFPS depending on minimizing the residual function. To do this, let us assume the solution v(x, t) of TF-FPEs (1) and (2) has the following MFPS expansion about By starting with the initial guess approximation v(x, 0) = v 0 (x), the series solution of equation (11) can be rewritten as To obtain the MFPS approximate solution, Now, define the kth-residual function as follows where the residual function can be given in the form Evidently, Res v (x, t) = 0 and lim where r is the radius of convergence for the MFPS (11). According to the RPSM, [25][26][27]30 it can be noted that These relations help us to determine the values of the coefficients v n (x), n = 1, 2, . . . , k. And so, the approximate solution for TF-FPEs (1) and (2) has been completely constructed. Anyhow, the next algorithm clarifies the procedure in obtaining the unknown coefficients of equation (13).
To determine the required coefficients of v k (x, t), do the following steps: Step 1: The initial condition v(x, 0) = v 0 (x), which is the zeroth FPS approximate solution of v(x, t).

Numerical experiments
The purpose of this section is to show the high degree of accuracy, efficiency, and applicability of this algorithm. The approximate analytical solutions of TF-FPEs are constructed in a rapidly convergent FPS form. Numeric comparisons of the results obtained by the proposed method, ADM 13 and VIM 13 are provided. The tabular and graphical results reveal that the RPS approach is easy to implement and accurate when applied to the TF-FPEs, as well as it introduces a promising tool for solving many fractional PDEs. The present computations are performed using Mathematica 10 (Wolfram Mathematica) software package.
Example 4.1. Consider the following TF-FPE with the initial conditions The exact solution of IVPs (17) and (18) for standard case at b = 1 is given by v(x, t) = xe t Using the last description of RPS algorithm, the solution of IVPs (17) and (18) is + 1))). The kth-residual function for equation (17) can be given by For k = 1, the first residual function is Depending on equation (16), the first unknown coefficient of MFPS expansion is v 1 (x) = x. Hence, the first FPS approximate solution is As the former, to determine the second coefficient v 2 (x), consider k = 2 in the MFPS (13) and then substitute v 2 (x, t) into the second residual function Res 2 v (x, t) of equation (19) to get By applying D b t on both sides of equation (22), it follows that Using the fact that Applying similar argument for k = 3, the third unknown coefficient in the MFPS (13) will be v 3 (x) = x. Moreover, the third FPS approximate solution can be written as In the same manner, the process can be repeated till the arbitrary order and then the coefficients of the MFPS solution (13) can be obtained. Consequently, we have v k (x) = x for k ø 1. Furthermore, if we collect all the last results, the solution v(x, t) can be given as follows where E g (t) = P ' j = 0 (t j =(G(gj + 1))) is the Mittag-Leffler function. 1 By setting b = 1, equation (26) can be reduced to v(x, t) = xe t , which is the exact solution of the classical form of IVPs (17) and (18).
In view of the obtained previous results and without loss of generality, the geometric behavior of the 10th FPS approximate solution of IVPs (17) and (18) has been studied by drawing the three-dimensional (3D) space figures at different values of b for x 2 ½0, 2 and t 2 ½0, 10.  Table 1. The results in Table 1 show that the FPS approximate solutions are in good agreement with exact solutions.
On the contrary, by applying the ADM, 13 we have the following iteration Using the property of equation (8), it follows that Continuing this process, the nth approximate solution is v n (x, t) = x(t nb =(G(nb + 1))) According to the ADM, 13 the kth ADM solution of IVPs (17) and (18) is given by Obviously, the RPSM produced an identical analytical solution of the ADM solution for this example. Anyhow, to see the effect of the fractional derivative to Fokker-Planck equation, the tabulated and graphical results for the approximate solutions at different values of fractional order b using the RPSM and (ADM) 13 with k = 20 are summarized and listed in Table 2 and Figure 2. It is obvious from the current results that the RPS algorithm is of good agreements with earlier literature works, and the RPS solution of TF-FPE approaches to the solution of the classical case as soon as b approaches to 1.

Example 4.2. Consider the following TF-FPE
t v x, t ð Þ= À ∂ ∂x ! v x, t ð Þ, with the initial conditions v x, 0 ð Þ= x 2 ð28Þ  The exact solution of IVPs (27) and (28) for standard case at b = 1 is given by v(x, t) = x 2 e t 2 In view of the RPS technique, by starting with v 0 (x) = x 2 as the initial approximation, the kth-residual function of IVPs (27) and (28) can be written as where v k (x, t) is the kth-truncated MFPS given by equation (13). Thus, for k = 1, the first residual function is given by Based on the result of equation (16), it yields that v 1 (x) = x 2 =2. Therefore, the first FPS approximate solution is To determine the second coefficient, let k = 2 in the kth-truncated MFPS (13), and substitute v 2 (x, t) into the Res 2 v (x, t) of equation (30) such that By considering the fact of equation (16) and solving D b t Res 2 v (x, 0) = 0 for v 2 (x), then it can obtain that v 2 (x) = x 2 =4. Hence, the second FPS approximate solution can be written as For the third unknown coefficient,  Now, compute D 2b t Res 3 v (x, t) for equation (34) and use D 2b t Res 3 v (x, 0) = 0 to get that v 3 (x) = x 2 =8. Therefore, the third FPS approximate solution is given by Using the same process for k ø 4, the kth unknown coefficient v k (x) can be obtained. Consequently, the solution v(x, t) of IVPs (27) and (28) can be expressed in the form of an infinite series given by For b = 1, the result in equation (36) can be reduced to v(x, t) = x 2 e t 2 , which represents the exact solution for classical form of TF-FPEs.
In view of the previous discussion, the geometric behavior of the 10th FPS approximate solution of IVPs (27) and (28) has been constructed and presented in To illustrate the efficiency and accuracy of the fractional residual power series (FRPS) algorithm, some numerical results at fixed value of x = 1 and some selected grid points t with step size 0:1 on ½0, 1 and k = 10 are given in Table 3.
On the contrary, by applying the ADM, 13 we have the following iteration Continuing this process, the kth-VIM solution can be given via Mathematica 10:0 for arbitrary k. Anyhow, Table 5 shows a comparison between approximate solutions using the RPSM, ADM, 13 and VIM 13 for different values of b, x, and t. As it is evident, the approximate solutions obtained by the RPSM and ADM are identical and are in a good agreement with the VIM approximate solutions for all values of b. It is to be noted that only the fourth term of each method is used in evaluating the results in Table 5.

Concluding remarks
Developing analytical and numerical solutions for fractional mathematical models of physical and chemical phenomena are very essential in science. In this work, an analytic-approximate method, so-called RPS, has been employed effectively to solve a class of Fokker-Planck PDEs of fractional order with fitted initial conditions. The RPS algorithm has been applied directly to obtain the solution in rapidly convergent MFPS without being linearized, discretized, or exposed to perturbation. Graphs and numerical results show that the proposed method is complete reliability and performance with great potential for use in many scientific applications. The present results show that the RPS technique is a simple and quite powerful tool in finding the approximate solutions for different kinds of fractional PDEs. A comparison between the RPSM and those available in the literature are carried out through numerical examples. High agreements of numerical results are clear and remarkable.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.