In this paper, Adomian decomposition method (ADM) and Modified ADM are used to obtain one soliton solution to the nonlinear telegraph equation. Some examples are provided to illustrate the method. The results show the simplicity and the efficiency of the method.

In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.

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In this paper, a variational homotopy perturbation method (VHPM) is based on variational iteration method and homotopy perturbation method, which is applied to solve nonlinear quadratic integral equations (QIEs), the (VHPM) find the solutions without any discretisation, transformation, or restrictive assumptions, some examples are given indicate the efficiency and simplicity of the proposed method.

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