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.