Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

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1 Commun. Theor. Phys. 69 ( Vol. 69, No. 5, May 1, 2018 Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations K. Parand, 1,2, S. Latifi, 1, M. M. Moayeri, 1, and M. Delkhosh 1, 1 Department of Computer Sciences, Shahid Beheshti University, G.C. Tehran, Iran 2 Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C. Tehran, Iran (Received October 9, 2017; revised manuscript received January 16, 2018 Abstract In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms. PACS numbers: Lj, Jn DOI: / /69/5/519 Key words: Fokker-Planck equations, Generalized Lagrange functions, Generalized Lagrange Jacobi Gauss- Lobatto (GLJGL collocation, Crank-Nicolson technique 1 Introduction In order to present the partial differential equation that is solved numerically, firstly, we give an introduction to the linear and nonlinear Fokker-Planck equations (FPEs and provide a brief review and history of these equations in the following subsection. 1.1 The Governing Equations The solution of the FPEs is important in various fields of natural science, including astrophysics problems, biological applications, chemical physics, polymer, circuit theory, dielectric relaxation, economics, electron relaxation in gases, nucleation, optical bistability, dynamics, quantum optics, reactive systems, solid-state physics, and numerous other applications. [1 The origin and history of FPEs go back to the time when Fokker-Planck described the Brownian motion of particles. [1 2 The theory of Brownian motion exists in many areas of physics and chemistry, and particularly in those that deal with the nature of metastable states and the rates at which these states decay. [3 Kramers equation is a special form of the FPEs utilized to describe the Brownian motion of a potential. [4 The general form of the FPEs, for the variable x and Corresponding author, k parand@sbu.ac.ir s.latifi@mail.sbu.ac.ir m moayeri@sbu.ac.ir mehdidelkhosh@yahoo.com c 2018 Chinese Physical Society and IOP Publishing Ltd t, is y [ t = 2 A(x + 2 B(x y, (1 y(x, 0 = f(x, x [0, 1, (2 where A(x and B(x are referred to as the drift and diffusion coefficients and in case the drift and diffusion coefficients depended on time we can show it as: y [ t = 2 A(x, t + 2 B(x, t y. (3 The above equation is considered as the equation of motion for the distribution function y(x, t, and is also called the forward Kolmogorov equation. In addition to the forward Kolmogorov equation, there is another form of the equation called backward Kolmogorov equation. y [ t = A(x, t 2 + B(x, t 2 y. (4 The more general forms of FPEs are its nonlinear form of the equation. The nonlinear FPEs may be derived from the principles of linear nonequilibrium thermodynamics. [5 Nonlinear FPEs have important applications and advantages in miscellaneous fields of sciences: biophysics, neurosciences, engineering, laser physics, nonlinear hydrodynamics, plasma physics, pattern formation, poly-

2 520 Communications in Theoretical Physics Vol. 69 mer physics, population dynamics, psychology, surface physics. [1,6 In the nonlinear FPEs, the equation also depends on y where this dependency happens in the drift and diffusion coefficients. The general form of this equation is y [ t = 2 A(x, t, y + 2 B(x, t, y y, (5 by which y t = A(x, t, yy } {{} drift term + 2 B(x, t, yy. (6 2 }{{} diffusion term Although there can be analytical solutions for the FPEs, it is difficult to result in solutions when the number of variables are large and no separation of variables methods are demanded. 1.2 The Literature Review on the FPEs In the early 1990s, Palleschi et al. [7 8 investigated FPEs. They discussed a fast and accurate algorithm for the numerical solution of Fokker-Planck-like equation. Vanaja [9 presented an iterative solution method for solving FPEs. Zorzano et al. [10 used the finite difference to investigate two-dimensional of this equation. Dehghan et al. [11 employed the He s variational iteration method (VIM to give a solution for this equation. Tatari et al. [12 applied the Adomian decomposition method for solving the FPEs. Using the cubic B-spline scaling functions, Lakestani et al. [2 obtained the numerical solution of FPEs. Kazem et al. [6 utilized RBF to solve the equation. Other insights for solving FPEs are numerical techniques. Among them, Wehner [13 applied path integrals to solve the nonlinear FPEs. Fourier transformations were employed by Brey et al. [14 Zhang et al. [15 applied distributed approximating functionals to solve the nonlinear FPEs. Further to these, for solving the one-dimensional nonlinear FPEs, the finite difference schemes [16 are also applied. In recent years, dozens of scientists are attracted to Spectral and pseudo spectral methods. [17 18 Spectral methods are providing the solution of the problem with the aid of truncated series of smooth global functions; [19 20 They provide such an accurate approximation for a smooth solution with relatively few degrees of freedom. They are widely employed in the approximation of the solution of differential equations, variational problems, and function approximation. The reason existed beyond this accuracy is that the spectral coefficients tend to zero faster than any algebraic power of their index n. [21 As said in such papers, spectral methods can fall into 3 categories: Collocation, Galerkin, and Tau methods [22 Collocation method provides highly accurate solutions to nonlinear differential equations. [23 26 There are only two main steps to approximate a problem in collocation methods: First, as a common approach, appropriate nodes (Gauss/Gauss-Radau/Gauss-Lobatto are chosen to represent a finite or discrete form of the differential equations. Second, a system of algebraic equations from the discretization of the original equation is obtained. [27 29 The Tau spectral method is one of the most important methods used to approximate numerical solutions of various differential equations. This method approximates the solution as an expansion of certain orthogonal polynomials/functions and the coefficients, in the expansion, are considered so as to satisfy the differential equation as accurately as possible. [30 Spectral Tau method is, somehow, similar to Galerkin methods in the way that the differential equation is enforced. [21 In Galerkin Spectral method, a finite dimensional subspace of the Hilbert space (trial function space are selected and trail and test functions are regarded the same. [31 Moreover, some numerical methods like Finite difference method (FDM and Finite element method (FEM that are implemented locally and require a network of data. Such methods like Meshfree methods do not require to build a network of data. [32 33 Comparing to these mentioned numerical methods, spectral methods are globally performing and they are continuous and do not need network construction. In addition to spectral methods, pseudospectral methods have been of high interest to authors presently. [34 37 Actually, in standard pseudospectral methods, interpolation operators are used to reduce the cost of computation of the inner product, in some spectral methods. For this purpose, a set of distinct interpolation points {x i } n i=0 is defined, where the corresponding Lagrange interpolants are achieved. In addition to this, in collocation points, {x i } n i=0, the residual function is set to vanish on the same set of points. Generally speaking, these collocation points do not need to be the same as the interpolation points; however, to have the Kronecker property, they are considered to be the same: therefore, by this trick, they reduce computational cost remarkably. [ The Main Aim of This Paper In this study, we develop an exponentially accurate generalized pseudospectral method for solving the linear and nonlinear FPEs: This method is a generalization of the classical Lagrange interpolation method. To reach this goal, in Sec. 2 some preliminaries of Jacobi polynomials are brought. In this section, we introduce the GL Functions and develop the GLJGL collocation scheme. Section 3 describes the numerical method; it explains the methodology and estimation of the error. We carry out numerical experiments to validate the presented collocation scheme. Subsequently, the analysis will be implemented to linear and nonlinear FPEs. Finally, some concluding remarks are given in Sec Preliminaries and Notations 2.1 Jacobi Polynomials The Jacobi polynomials are the eigenfunctions of a singular Sturm-Liouville equation. There are several

3 No. 5 Communications in Theoretical Physics 521 particular cases of them, such as Legendre, the four kinds of Chebyshev, and Gegenbauer polynomials. Jacobi polynomials are defined on [ 1, 1 and are of interest recently. [36,40 43 The recursive formula for Jacobi polynomials is as follows: [44 P α,β (α + i 1(β + i 1(α + β + 2i i (x = P α,β i 2 i(α + β + i(α + β + 2i 2 (x + (α+β+2i 1{α2 β 2 + x(α+β+2i(α +β + 2i 2} 2i(α + β + i(α + β + 2i 2 P α,β i 1 (x, i = 2, 3,..., (7 where P α,β 0 (x = 1, P α,β 1 (x = α + β + 2 x + α β, 2 2 with the properties as: Pn α,β ( x = ( 1 n Pn β,α (x, (8 Pn α,β ( 1 = ( 1n Γ(n + β + 1, n!γ(β + 1 (9 P α,β n (1 = Γ(n + α + 1 n!γ(α + 1, (10 (Pn α,β (x (m m Γ(m + n + α + β + 1 = 2 Γ(n + α + β + 1 P α+m,β+m n m (x, (11 and its weight function is w α,β (x = (1 x α (1 + x β. Moreover, the Jacobi polynomials are orthogonal on [ 1, 1: γn α,β 2 α+β+1 Γ(n + α + 1Γ(n + β + 1 = (2n + α + β + 1Γ(n + 1Γ(n + α + β + 1, 1 1 P α,β n (xpm α,β (xw α,β (x = δ m,n γn α,β, (12 where δ m,n is the Kronecker delta function. The set of Jacobi polynomials makes a complete L 2 w [ 1, 1 orthogonal system for any g(x α,β [ 1, 1, there is an expansion as follows. L 2 w α,β L where g(x = j=0 b j P α,β j (x, b j = 1 1 γ α,β g(xp α,β j (xw α,β (xdx. j Generalized Lagrange (GL functions In this section, generally, the GL functions are introduced and suitable formulas for the first- and second-order derivative matrices of these functions are presented. Definition 1 Considering N w(x = (u(x u(x i i=0 the generalized Lagrange (GL functions formula can be shown as: [38 39 L u j (x = w(x (u u j x w(x j = u j w(x (u u j u w(x j w(x = κ j, j = 0,..., N, (13 (u u j where κ j = u j / uw(x j, u w(x = (1/u x w(x, and u(x is a continuous and sufficiently differentiable function which will be chosen to fit in the problem s characteristics. For simplicity u = u(x and u i = u(x i are considered. The GL functions have the Kronecker property: { 0 if k j, L u j (x k = δ j,k = 1 if k = j. Theorem 1 Considering the GL functions L u j (x in Eq. (13, one can exhibit the first-order derivative matrices of GL functions as D (1 = [d kj 0 j,k n R (n+1 (n+1, where x w(x j κ j, j k, u k u j d kj = u j κ 2 xw(x j u j xw(x j j 2, j = k. 2u j Proof As the GL functions defined in Eq. (13, the firstorder derivative formula for the case k j can be achieved as follows: d kj = x L u L u j j (x k = lim (x Lu j (x k x x k x x k x w(x k = κ j. (14 u k u j But, when k = j, with L Hôpital s rule: d jj = x L u j (x j = lim x x j κ j (u u j x w(x u j w(x (u u j 2 H = lim x x j κ j (u u j 2 xw(x u w(x 2u (u u j H u j = κ 2 xw(x j u j xw(x j j 2u 2. j This completes the proof. 2.3 Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL Collocation Method In case of GLJGL collocation method, w(x in Eq. (13 can be restated as: w(x = λ(u u 0 (u u n P α+1,β+1 n 1 (u, where λ is a real constant and to simplify the notation, we write J(u = P α+1,β+1 n 1 (u, (15 with the following important properties: x J(u = u Γ(α + β + n + 3 2Γ(α + β + n + 2 P α+2,β+2 n 2 (u, (16

4 522 Communications in Theoretical Physics Vol. 69 Then, we have: J(u 2 Γ(α + β + n + 3 = u (2 u 4Γ(α + β + n + 2 u P α+2,β+2 n 2 (u + u (α + β + n + 3P α+3,β+3 n 3 (u, (17 J(u 0 = ( 1n 1 Γ(β n (n 1!Γ(β + 2, J(u n = Γ(α n (n 1!Γ(α + 2. (18 x w(x = λ[u (u u n J(u + u (u u 0 J(u + (u u 0 (u u n x J(u, (19 2 xw(x = λ[u (u u n J(u + 2(u 2 J(u + 2u (u u n x J(u + u (u u 0 J(u + 2u (u u 0 x J(u + (u u 0 (u u n 2 xj(u. (20 Recalling that {P α+1,β+1 n 1 (u j = 0} 1 n 1 and using formulas in Eq. (15 (20, we find the entry of the first-order derivative matrix of GL functions as: u k P α+2,β+2 n 2 (u k (u k u n (n 1!(α + β + n + 2Γ(β + 2 2(u 0 u n ( 1 n 1, j = 0, 1 k n 1, Γ(β n u nγ(α + n + 1Γ(β + 2 (u 0 u n ( 1 n 1 Γ(β + n + 1Γ(α + 2, j = 0, k = n, 2( 1 n 1 u 0(u 0 u n Γ(β + n + 1 Γ(β + 2(n 1!(α + β + n + 2P α+2,β+2 n 2 (u j (u 0 u j 2 (u j u n, k = 0, 1 j n 1, u 0Γ(β + n + 1Γ(α + 2( 1 n 1 (u n u 0 Γ(α + n + 1Γ(β + 2, k = 0, j = n, d kj = u k P α+2,β+2 n 2 (u k (u k u n (n 1!(α + β + n + 2Γ(α + 2, j = n, 1 k n 1, 2(u n u 0 Γ(α n (u k u 0 (u k u n u k P α+2,β+2 n 2 (u k (u j u 0 (u j u n (u k u j P α+2,β+2 n 2 (u j, 1 j k n 1, 2u n(u n u 0 Γ(α + n + 1 (u j u 0 (u j u n 2 P α+2,β+2 n 2 (u j (n + α + β + 2(n 1!Γ(α + 2, 1 j n 1, k = n, u j u j u 0 + u j + u u j u j(α + β + n + 3 P α+3,β+3 n n 3 (u j 4P α+2,β+2 n 2 (u j, 1 k = j n 1, u j + u j + u α+2,β+2 j (n + α + β + 2Pn 2 (u j 2u j u j u n 2P α+1,β+1, k = j = 0, or k = j = n. n 1 (u j Theorem 2 Let D (1 be the above matrix (first order derivative matrix of GL functions and define matrix Q such that Q = Diag (u 0, u 1,..., u N, Q(1 = Diag (u 0, u 1,..., u N, then, the second-order derivative matrix of GL functions can be formulated as: Proof See Ref. [38. D (2 = (Q (1 + QD (1 Q 1 D (1. (21 3 Numerical Method In this section, firstly, the time discretization method is recalled. Secondly, GLJGL collocation method is implemented to solve the FPEs. In a matrix form, the method has been presented and the error of this method is estimated. 3.1 Discretization y i+1 (x y i (x t = 1 2 For solving the FPEs, we first discretize the time domain; to do this, we apply the Crank-Nicolson method. The main reason for choosing this method is its good convergence order and its unconditional stability. [45 To apply this method, firstly, we approximate and simplify the first-order derivative of y(x, t, with respect to the time variable, and deriving a formula from finite difference approximations as follows: y(x, t y(x, t + t y(x, t. (22 t t The domain Ω [0, T is decomposed as Ω [0, T = Ω s 1 i=0 [T i, T i+1, where T 0 = 0, T s = T, T i+1 T i = t, and t = T/s: The error of this approximation is of order O( t. From now on, for simplicity y i (x = y(x, T i. Considering FPEs, one can read y(x, t y(x, t 2 y(x, t = E 0,k y(x, t + E 1,k + E 2,k t 2, (23 in which E 0,k, E 1,i and E 2,i are the coefficient specified in the Numerical Examples section; in linear FPEs, E 0,i = 0. Implementing Crank-Nicolson on FPEs ( E 0,k y i+1 y i+1 (x 2 y i+1 (x (x + E 1,k + E 2,k 2

5 No. 5 Communications in Theoretical Physics 523 and can be simplified as ( E 0,k y i y i (x 2 y i (x (x + E 1,k + E 2,k 2, (24 y i+1 (x t ( E 0,k y i+1 y i+1 (x (x + E 1,k 2 = y i (x + t 2 2 y i+1 (x + E 2,k 2 ( E 0,k y i y i (x 2 y i (x (x + E 1,k + E 2,k 2. (25 By applying this method, the problem can be discretized in small time levels. As shown, time variable is discretized using Crank-Nicolson method. In each time level, we are to approximate the FPEs. Solving in sufficiently large time levels, brings in a good approximation for FPEs. 3.2 Implementation of GLJGL Collocation Method for Solving FPEs As said in the previous subsection, in each time level, we approximate the solution of FPEs, and therefore, the time variable is omitted from the equation. In each time step, we approximate an equation like in Eq. (25. The unknown y i+1 (x is approximated as where y i+1 n (x = LA i+1, (26 A i+1 = [a i+1 0,..., a i+1 n T, L = [L u 0(x,..., L u n(x. As y(x, 0 = y 0 (x = f(x we can calculate f(x = LA 0, and by collocating n + 1 nodes we can result in: A 0 = [f(x 0,..., f(x n T. (27 By the aid of these, we can write Eq. (25 as LA i+1 t 2 (E 0,kLA i+1 + E 1,k L A i+1 + E 2,k L A i+1 = LA i + t 2 (E 0,kLA i + E 1,k L A i + E 2,k L A i. (28 The boundary conditions, by considering Guass-Lobatto scheme and Eq. (26, are specified as: y i+1 (x 0 = a i+1 0 = y(0, T i+1, y i+1 (x n = a i+1 n = y(1, T i+1, (29 therefore, by collocating n + 1 points and defining H 0 = diag (0, E 0,1, E 0,2,..., E 0,n 2, E 0,n 1, 0, H 1 = diag (0, E 1,1, E 1,2,..., E 1,n 2, E 1,n 1, 0, H 2 = diag (0, E 2,1, E 2,2,..., E 2,n 2, E 2,n 1, 0, M = t 2 (H 0 + H 1 D + H 2 D (2, R = [y(0, T i+1 a i 0, 0,..., 0, y(1, T i+1 a i n T, then, the matrix form of Eqs. (28 and (29 will be ( I M A i+1 = ( I + M A i + R. (30 The first and last row of matrices H 0, H 1, H 2, and first and last elements of vector R are defined as if they satisfy the boundary condition of FPEs. Hence, we can achieve the numerical solution of y(x, t at each time level. Notice that, at time level 0 the solution is computed from the initial condition; This is shown in Eq. (27. From the solution of the system in Eq. (30, at each time level, for the next time levels, we will achieve the unknown values. In other words, it means that by solving this system, in each step of i + 1, the unknown coefficients A i+1 will be found. This system of equations is solved by applying a proper method like Newton methods. To show the accuracy of this method, some examples in the next section, are illustrated. 3.3 Error Estimation Theorem 3 Let x 0 = a, x n = b and {x i } n 1 i=1 be the roots of P α,β n 1 (u(x, where u(x = 2(x a/(b a 1 shifting (x from [ 1, 1 to [a, b. Then, there exists a unique set of quadrature weights {ϖ} n i=0 defined by Jie Shen [46 (Jacobi Gauss-Lobatto quadratures, such that for all functions p(x of degree 2n 1 Jacobi polynomial P α,β n 1 b a p(xw(x = n p(x i ϖ i, (31 i=0 where w(x is the weight function and here this weight function is w α,β( u(x. This is worth noticing that x i = b a 2 t i + b + a, ϖ i = b a ˆϖ i, 2 2 {t i, ˆϖ i } n i=0 are Jacobi Gauss-Lobatto quadratures nodes and weights. Proof See Ref. [46. In FPEs [a, b = [0, 1, u(x = 2x 1, then 1 n p(xw(x = p(x i ϖ i, (32 0 i=0 based in the last theorem, when p(x P m, m > 2n 1, the above relation between integral and summation is not exact; it produces an error term as er [p(x = 1 (n + 1! where ξ (a, b. Hence, b a p(xw(x = b a p (n+1 (ξ n (x x i dx, i=0 n p(x i ϖ i + er [p(x. (33 i=0 For two arbitrary functions g 1 (x and g 2 (x we define g 1 (x, g 2 (x = b a g 1 (xg 2 (xw(xdx, then for L u i Lu i P 2n we have n L u i, L u i = L u i (x k L u i (x k ϖ k + er [L u i (xl u i (x. k=0

6 524 Communications in Theoretical Physics Vol. 69 In the same fashion, for L u i (Lu i P 2n 1 and L u i (Lu i P 2n 2 L u i, (L u i = n L u i (x k (L u i (x k ϖ k, k=0 y i+1 (x, L u k t ( E 0,k y i+1 (x, L u y i+1 (x 2 k + E 1,k = y i (x, L u k + t ( E 0,k y i (x, L u y i (x 2 k + E 1,k L u i, (L u i = n L u i (x k (L u i (x k ϖ k. k=0 Now, by multiplying Eq. (25 with L u k (xwα,β (x and integration in both sides:, L u k + + E 2,k 2 y i+1 (x 2, 2 y i (x Lu k E 2,k 2, L u k With Eqs. (26 and (33 the following relations in x k will be obtained: (j = i, i + 1 y j (x k, L u k(x k = ϖ k I[k, :A j + er [y j (x k L u k(x k,, L u k. (34 E 0,k y j (x k, L u k(x k = E 0,k ϖ k I[k, :A j + er [E 0,k y j (x k L u k(x k, y j (x k [ E 1,k, L u k(x k = E 1,k ϖ k D[k, :A j y j (x k + er E 1,k L u k(x k, 2 y j (x k [ E 2,k 2, L u k(x k = E 2,k ϖ k D (2 [k, :A j 2 y j (x k + er E 2,k 2 L u k(x k, in which D[k, : means that the k-th row of matrix D is taken. Now, by taking x k into account. k = 0,..., n ( I M A i+1 = ( I + M A i + R + V. (35 Comparing with the system in Eq. (30 we solved, V is the error term vector: V is defined as: v k = 1 [ er y i (xl u ϖ k(x + t k 2 E 0,ky i (xl u k(x + t 2 E y i (x 1,k Lu k(x + t 2 E 2 y i (x 2,k 2 L u k(x y i+1 (xl u k(x t 2 E 0,ky i+1 (xl u k(x t 2 E 1,k y i+1 (x L u k(x t 2 E 2,k 2 y i+1 (x 2 L u k(x x=x k for k = 1,..., n 1, and v 0 = 0, v n = 0. As er [q(x = 0, as long as q(x P m, m 2n 1. Obviously, if any of the above terms degree is less and equal than 2n 1, the error of that term will be zero. In numerical examples, this error is shown and discussed. 4 Numerical Examples In this section, in order to illustrate the performance of the GLJGL collocation method, we give some computations based on preceding sections, to support our theoretical discussion. By the aid of the presented method, linear and nonlinear forms of FPEs are solved. To illustrate the good accuracy of these methods, we apply different error criteria: The root-mean-square (RMS, N e, and L 2 errors. r j=1 RMS = (y(x j y n (x j 2, r r j=1 N e = (y(x j y n (x j 2 r, j=1 y2 (x j r L 2 = (y(x j y n (x j 2, j=1 where y(x j and y n (x j are exact and approximate value of FPEs on equidistant x j, j = 1,..., r. As FPEs are defined over [0, 1, the shifting function u(x, considered in Subsecs. 2.2 and 2.3, is u(x = 2x 1. The CPU time for calculation of matrices D (1 and D (2, defined in Subsec. 2.3, is brought in Table 1. Table 1 CPU time (sec for calculation of derivative matrices for different values of n. n CPU time of D (1 CPU time of D ( The CPU time is performed on a DELL laptop with the configuration: Intel(R Core(TM i7-2670qm CPU, 2.20 GHz; and 6 GB RAM. Example 1 Consider Refs. [2, 6, 11 Eq. (1 with: A(x = 1, B(x = 1, f(x = x, x [0, 1. The exact solution of this test problem is y(x, t = x+t. In this example E 0,k = 0, E 1,k = A(x k = 1, and E 2,k = B(x k = 1 for k = 1,..., n 1. As stated earlier, if the order of terms in Eq. (35 is less than 2n, the error terms vanish; so, the error vector for Ex. 1, V in Eq. (35, can be simplified as v k = 1 ϖ k er [y i (xl u k(x y i+1 (xl u k(x x=xk, k = 1,..., n 1, v 0 = 0, v n = 0. In Table 2, the numerical absolute errors of Example 1, and their comparison with B-Spline method are displayed. Table 3, by representing the values of RMS and N e errors,

7 No. 5 Communications in Theoretical Physics 525 reveals the difference between the presented method and both HRBF and Kansa s approaches. [6 In Fig. 1, RMS, L 2 and N e errors, for different values of n and t, have been illustrated. Figure 2 shows the plot of error for Ex. 1. Table 2 Numerical absolute errors of the method for Ex. 1, in comparison with B-Spline method. [2 n = 20, t = 0.01, α = 0, β = 1. t x = 0.2 x = 0.4 x = 0.6 x = 0.8 B-Spline Presented B-Spline Presented B-Spline Presented B-Spline Presented method method method method Fig. 1 Plot of results for Ex. 1, α = 0, β = 1, r = 20. (a Value of error measurements for different values of t. n = 20 is fixed; (b Value of error measurements for different values of n. t = 0.01 is fixed. Fig. 2 Plot of absolute error of Ex. 1, α = 0, β = 1, r = 20, t = 0.01, n = 20. Example 2 Consider Refs. [2, 6, 11 the backward Kolmogorov Eq. (4 with: A(x, t = (x+1, B(x, t = x 2 e t, f(x = x + 1, x [0, 1. The exact solution of this test problem is y(x, t = (x + 1 e t. In this example E 0,k = 0, E 1,k = A(x k, t = (x k + 1, and E 2,k = B(x k, t = x 2 k et for k = 1,..., n 1. Table 4 depicts the numerical absolute errors of Ex. 2 and draws a distinction with the presented method and B- Spline method. For showing the accuracy, the differences between the presented method and HRBF and Kansa s approaches [6 are shown by calculating RMS and N e in Table 5. In Fig. 3, the error measurements RMS, L 2 and N e are shown for different n and t. In this figure, CPU times have been depicted for different n and t. It explicitly says that when n increases or t decreases, the time of solving the system of Eq. (30 increases. As it shows, when t tends to a smaller value, it affects and decreases all RMS, N e, L 2 and absolute errors. The plot of absolute error for Ex. 2 is also shown in Fig. 4.

8 526 Communications in Theoretical Physics Vol. 69 Table 3 Values of RMS and N e for Ex. 1 in comparison with HRBF and Kansa s approaches. r = 20, t = GLJGL-c (n = 15 HRBF without time-discretization [6 Kansa s approach [6 α β RMS N e n RMS N e RMS N e Table 4 Numerical absolute errors of the method for Ex. 2, in comparison with B-Spline method. [2 n = 20, t = 0.01, α = 0, β = 1. t x = 0.2 x = 0.4 x = 0.6 x = 0.8 B-Spline Presented B-Spline Presented B-Spline Presented B-Spline Presented method method method method Fig. 3 Plot illustration results of Ex. 2, α = 0, β = 1, r = 20. (a CPU times for solving Eq. (30 for different values of t and n. (b Value of error measurements for different values of t. n = 20 is fixed. (c Plot of absolute error for different values of t. n = 20 is fixed. (d Value of error measurements for different values of n. t = 0.01 is fixed.

9 No. 5 Communications in Theoretical Physics 527 y t = ( 3y + 2xy x y x + (2xyy xx. (36 By Eqs. (23 and (36 one can set: E 1,k = 0, E 1,k = 3y(x k + 2x k y x (x k, E 2,k = 2x k y(x k, k = 1,..., n 1. Fig. 4 Plot of absolute error of Ex. 2 for 15 collocation points. α = 0, β = 1, t = Example 3 Consider Refs. [2, 6, 11 the nonlinear Eq. (5 with: A(x, t, y = (7/2y, B(x, t, y = xy, f(x = x, x [0, 1. The exact solution of this test problem is y(x, t = x/(1 + t. By this consideration, Eq. (5 can be rewritten as y t = 3yy x + 2xy 2 x + 2xyy xx, The error vector, V in Eq. (35, for Ex. 2 and 3 is v k = 1 [ er y i (xl u ϖ k(x + t k 2 E y i (x 1,k Lu k(x + t 2 E 2 y i (x 2,k 2 L u k(x y i+1 (xl u k(x t 2 E y i+1 (x 1,k L u k(x t 2 E 2 y i+1 (x 2,k 2 L u k(x, x=x k k = 1,..., n 1, v 0 = 0, v n = 0. By the aid of Table 6. the numerical absolute errors for Ex. 3 demonstrated and a comparison with the B-Spline method is made. For this example, also, RMS and N e are compared with the ones provided by HRBF [6 in Table 7. Table 5 Values of RMS and N e for Ex. 2 in comparison with HRBF and Kansa s approaches. r = 50, t = GLJGL-c (n = 10 HRBF without time-discretization [6 Kansa s approach [6 α β RMS N e n RMS N e RMS N e Table 6 Numerical absolute errors of the method for Ex. 3, in comparison with B-Spline method. [2 n = 10, t = 0.001, α = 1, β = 1. t x = 0.2 x = 0.4 x = 0.6 x = 0.8 B-Spline Presented B-Spline Presented B-Spline Presented B-Spline Presented method method method method Table 7 Values of RMS and N e for Ex. 3 in comparison with HRBF approach. r = 50, t = GLJGL-c (n = 7 HRBF without time-discretization [6 HRBF with time-discretization [6 α β RMS N e n RMS N e n RMS N e

10 528 Communications in Theoretical Physics Vol. 69 Fig. 5 Plot illustration results of Ex. 3, α = 1, β = 1, r = 50. (a CPU times for solving Eq. (30 for different values of t and n. (b Value of error measurements for different values of t. n = 10 is fixed. (c Plot of absolute error for different values of t. n = 10 is fixed. (d Value of error measurements for different values of n. t = is fixed. Example 4 Consider Refs. [2, 6, 11 the nonlinear Eq. (5 with: A(x, t, y = 4(y/x x/3, B(x, t, y = y, f(x = x 2, x [0, 1. The exact solution of this test problem is y(x, t = x 2 e t. This nonlinear FPEs can be restated as ( x 2 y 8 ( x x y x + 2y xx y y x y x. (37 It must be noted that: the way this relation is factorized is playing a central role in the exactness of solution. By Eqs. (23 and (37: E 0,k = x 2 k y(x k 8 x k y x (x k + 2y xx (x k, Fig. 6 Plot of absolute error of Ex. 3 for 7 collocation points. α = 1, β = 1, t = Figure 5 shows the values of RMS, L 2 and N e errors for different n and t. This Figure, illustrates the CPU times for solving the system of Eq. (30 for different n and t. It shows that when n increases or t decreases, the time of obtaining solution will increase. The fact is, as t becomes smaller, RMS, Ne, L 2 and absolute errors decrease. The plot of absolute error for Ex. 3 is also shown in Fig. 6. E 1,k = x k 3 + 2y x(x k, E 2,k = 0, k = 1,..., n 1. For Ex. 4, the error vector specified in Eq. (35 is v k = 1 [ er ϖ k y i (xl u k(x + t 2 E 0,ky i (xl u k(x + t 2 E y i (x 1,k Lu k(x y i+1 (xl u k(x t 2 E 0,ky i+1 (xl u k(x t 2 E y i+1 (x 1,k L u k(x x=x k for k = 1,..., n 1 and v 0 = 0, v n = 0.

11 No. 5 Communications in Theoretical Physics 529 In Table 8, the numerical absolute errors for Ex. 4 demonstrated and a comparison with the B-Spline method is given. The error measurements RMS and N e are calculated by the presented method and HRBF [6 method and the results depicted in Table 9. Figure 7 illustrates the values of RMS, L 2 and N e errors for different n and t. This Figure, also, illustrates the CPU times for solving the system of Eq. (30 for different n and t. It implies that when n increases or t decreases, the time of obtaining solution increases. In fact, when t becomes smaller, RMS, Ne, L 2 and absolute errors will decrease. The plot of absolute error for Ex. 4 is also shown in Fig. 8. Fig. 7 Plot illustration results of Ex. 4, α = 1, β = 1, r = 50. (a CPU times for solving Eq. (30 for different values of t and n. (b Value of error measurements for different values of t. n = 7 is fixed. (c Plot of absolute error for different values of t. n = 7 is fixed. (d Value of error measurements for different values of n. t = is fixed. Table 8 Numerical absolute errors of the method for Ex. 4, in comparison with B-Spline method. [2 n = 7, t = 0.001, α = 1, β = 1. x = 0.2 x = 0.4 x = 0.6 x = 0.8 t B-Spline Presented B-Spline Presented B-Spline Presented B-Spline Presented method method method method Table 9 Values of RMS and N e for Ex. 4 in comparison with HRBF approach. r = 50, t = GLJGL-c (n = 5 HRBF without time-discretization [6 HRBF with time-discretization [6 α β RMS N e n RMS N e n RMS N e

12 530 Communications in Theoretical Physics Vol. 69 Fig. 8 Plot of absolute error of Ex. 4, α = 1, β = 1, t = 0.001, n = 7. 5 Conclusion The (linear and nonlinear FPEs have many applications in science and engineering. So, in this work, a numerical method based on GLJGL collocation method is discussed and developed to investigate FPEs. Firstly, we introduced GL functions with the Kronecker property. The advantages of using GL functions can be: (i These functions are the generalization of the classical Lagrange polynomials and corresponding differentiation matrices of D (1 and D (2, as shown, can be reached by specific formulas; this helps create and introduce a derivative-free method. (ii With different consideration of u(x, different basis of GL functions are provided; therefore, different problems defined on various intervals can be solved. (iii The accuracy of the presented method by GL function has exponential convergence rate. Moreover, the time derivative of the FPEs is discretized using Crank-Nicolson method. The main reason for using Crank-Nicolson method is its unconditional stability. [3,45 By the aid of Crank-Nicolson technique, we solved the linear and nonlinear types of FPEs with GLJGL collocation method. We apply the pseudospectral method in a matrix based manner where the matrix based structure of the present method makes it easy to implement. Also, to show the accuracy and ability of the proposed method, several examples are solved. Several examples are given and the results obtained using the method introduced in this article show that the new proposed numerical procedure is efficient The results showed that the approximate solutions of the GLJGL collocation method can be acceptable and provides very accurate results even with using a small number of collocation points. To illustrate the suitable accuracy of the proposed method, we used three different error criteria, namely, RMS, L 2 and N e. Additionally, the obtained results have been compared with B-Spline, HRBF and Kansa methods, showing the accuracy and reliability of the presented method. This method can also be used as a powerful tool for investigation of other problems. References [1 H. Risken, The Fokker-Planck Equation: Method of Solution and Applications, Springer Verlag, Belin, Heidelberg (1989. [2 M. Lakestani and M. Dehghan, Numer. Method. Part. D. E 25 ( [3 M. Dehghan and V. Mohammadi, Eng. Anal. Bound. Elem. 47 ( [4 S. Jenks, Introduction to Kramers Equation, Drexel University, Philadelphia (2006. [5 A Compte and D Jou, J. Phys. A-Math. Gen. 29 ( [6 S. Kazem, J. A. Rad, and K. Parand, Eng. Anal. Bound. Elem. 36 ( [7 V. Palleschi, F. Sarri, G. Marcozzi, and M. R. Torquati, Phys. Lett. A 146 ( [8 V. Palleschi and N. de Rosa, Phys. Lett. A 163 ( [9 V. Vanaja, Appl. Numer. Math. 9 ( [10 M. P. Zorzano, H. Mais, and L. Vazquez, Appl. Math. Comput. 98 ( [11 M. Dehghan and M. Tatari, Physica Scripta 74 ( [12 M. Tatari, M. Dehghan, and M. Razzaghi, Math. Comput. Model. 45 ( [13 M. F. Wehner and W. G. Wolfer, Phys. Rev. A 35 ( [14 J. J. Brey, J. M. Casado, and M. Morillo, Phys. A 128 ( [15 D. S. Zhang, G. W. Wei, D. J. Kouri, and D. K. Hoffman, Phys. Rev. E 56 ( [16 A. N. Drozdov and M. Morillo, Phys. Rev. E 54 ( [17 A. H. Bhrawy, M. A. Abdelkawy, J. T. Machado, and A. Z. M. Amin, Comput. Math. Appl. 2016:doi. org/ /j.camwa [18 A. H. Bhrawy, Numer. Algorithm. 73 ( [19 K. Parand and M. Delkhosh, J. Comput. Appl. Math. 317 ( [20 K. Parand and M. Delkhosh, Boletim da Sociedade Paranaense de Matemática 36 ( [21 A. H. Bhrawy and M. M. Al-Shomrani, Adv. Differ. E 2012 ( [22 E. H. Doha and A. H. Bhrawy, Appl. Numer. Math. 58 ( [23 A. H. Bhrawy and M. M. Alghamdi, Boundary Value Prob (

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