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In this work, we aim to apply a numerical approach based on Homotopy perturbation transform method (HPTM) for derive the exact and approximate solutions of nonlinear fifth order KdV equations for study magneto-acoustic waves in plasma. The approach is a mixed form of the standard Laplace transform with the classical Homotopy perturbation technique. Nonlinear term can be handled with the aid of He's Polynomials. In this technique, the solution is calculated in the form of a convergent series and convergence of the HPTM solutions to the exact solutions is shown. The Homotopy perturbation transform method presents a wide applicability to handling nonlinear wave equations in science and engineering. As a novel application of HPTM, the present work shows some essential difference with the existing similar method. Several examples are provided for illustrate the simplicity and reliability of the method and highlighted the significance of this work. Keywords: Laplace transform, Homotopy perturbation method, Fifth order KdV equation, Non-homogeneous KdV equation, He's Polynomials

The objective of this paper is to present a reliable approach to compute an approximate analytical solution of magneto-hydrodynamic (MHD) Jeffery–Hamel flow by using a new modification of Adomian decomposition method. The approximate solution of this problem is calculated here in the form of a rapidly convergent series at one and both grid points. Results for velocity profiles in divergent and convergent channels are presented for different values of Hartmann and Reynolds numbers. The validity and applicability of this new technique are illustrated through a comparison between reproducing Kernel Hilbert Space Method (RKHSM) (Inc et al., 2013), Modified Homotopy Perturbation Method (MHPM) (Singh, 2014) and numerical solutions. Keywords: Jeffery–Hamel flow, Modified Adomian Decomposition Method (MADM), Nonlinear differential equation

The main aim of the present paper was to present a user friendly approach based on homotopy analysis transform method to solve a time-fractional nonlinear shock wave equation arising in the flow of gases. The proposed technique presents a procedure of constructing the set of base functions and gives the high-order deformation equations in a simple form. The auxiliary parameter ℏ in the homotopy analysis transform method solutions has provided a convenient way of controlling the convergence region of series solutions. The method is not limited to the small parameter, such as in the classical perturbation method. The technique gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive.

In this paper we will use the residual power series (RPS) method to generate the solution of the nonlinear time fractional Navier-Stokes equation in two dimensions in the form of a rapidly convergent series, where we adopted the definition of Caputo to the fractional differential derivative of time. We gave an illustrative application to demonstrate the effectiveness and leverage of the proposed method, where numerical results and graphical representations reveal that the proposed method performs extremely well in terms of efficiency and simplicity and it can be utilized to solve more problems in the field of non-linear fractional differential equations (FDEs). Keywords: Fractional Navier-Stokes equation, Caputo's fractional derivative, Power series solution

Differential algebraic equations arise in many applications. In this paper, the homotopy perturbation method is applied to solve differential algebraic equations. In this method, the solution is found in the form of a convergent series and usually converges to the exact solution. In addition, the terms of the series can be computed easily. A few examples are presented to illustrate the method. The results show the simplicity and efficiency of the proposed method.