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Since achieving the precise analytical solution of most nonlinear problems is not possible, numerical methods are meant to be used. In fact a very limited number of nonlinear problems have the exact analytical solution. Let us consider some other methods which are known as semi-analytical methods and applied to solve nonlinear problems. In these methods by using a primary approximation and an iterative process, the approximate solution can be provided, so that the iterations converge to the desired solution. Methods which have been employed and support researchers in this field are categorized as "Adomian decomposition method"(ADM), "Homotopy perturbation method"(HPM), "Homotopy analysis method"(HAM), and "He variational iteration method"(VIM). The aim of this paper is to suggest an improvement of reliability of Adomian decomposition method to obtain a solution to Duffing equation. In details, Adomian decomposition method using modified Legendre polynomials has been applied in the present work that allows researchers to provide approximation of the solution with high accuracy. This method is tested for example. Moreover the obtained numerical results will be presented.

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

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