In this work, we have generalized the homotopy perturbation method (HPM) and we have shown that methods such as the spectral method, the Adomian decomposition method and the HPM are special cases of the new modified HPM of the present study. At last we apply the HPM to solve the Fokker–Planck equation. To illustrate the method some experiments are provided. The results reveal the efficiency and accuracy of the HPM. The HPM can obtain analytic form of the solution in some cases. The technique (HPM) has been applied with great success to obtain the solution of a large variety of nonlinear problems in both ordinary and partial differential equations, integral equations and integro-differential equations.

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Abstract. In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, error estimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method.Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

In this paper, a technique generally known as meshless method is presented for solving fractional partial differential equations (FPDEs). Some physical linear and nonlinear experiments such as time-fractional convective-diffusion equation, timefractional wave equation and nonlinear space-fractional Fisher's equation are considered. We present the advantages of using the radial basis functions (RBFs) especially wherein the data points are scattered. Comparing between the numerical results obtained from our method and the other methods confirms the good accuracy of the presented scheme.

a b s t r a c tIn this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane-Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane-Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Padé series form, we obtain an approximate polynomial of arbitrary order for solving Lane-Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

In this paper, the homotopy perturbation method (HPM) is improved to solve nonlinear diffusion equations. Nonlinear diffusion equations are the most important equations in mathematical physics. By applying this improvement (IHPM), exact solutions are obtained for some important physical processes of power-law diffusivities. To illustrate the method, some examples are provided. The convergence of the method is also considered. The results show the efficiency and accuracy of the IHPM.

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The nonlinear Burgers' equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers' equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers' equation. These methods are based on the multiquadric (MQ) quasi-interpolation operator L W 2 and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers' equation.