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Abstract:In this research article, we give analytic approximate solution to the Sharma Tasso Olver (STO) equation and exact solutions to both the Schrodinger equation and the Telegraph equation. Also, the approximate analytical and exact solutions we present in this paper are calculated in the form of power series with easily computable components. The obtained results are in a good agreement with the exact solutions. We present an algorithm called the Reduced Differential Transform Method (RDTM) to find approximate solution and we compare the results with the exact solutions. This method reduces significantly the numerical computations compare with the existing methods such as the perturbation technique, differential transform method (DTM) and the Adomian decomposition method (ADM).

In this study, we present convergence analysis along with an error estimate for time-fractional biological population equation in terms of the Caputo derivative using a new technique called the fractional decomposition method (FDM). Further, we present exact solutions to four test problems of nonlinear time-fractional biological population models to show the accuracy and efficiency of the FDM. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without the need of linearization, discretization and perturbations. The results prove that the FDM is very effective and simple for solving fractional partial differential equations in multi-dimensional spaces, special cases of which we have described in this paper.

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In shallow waters, the Wu-Zhang (WZ) system describes the (1+1)-dimensional dispersive long wave in two horizontal directions, which is important for the engineering community. This paper presents proofs for various theorems and shows that the natural decomposition method (NDM) solves systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions, such as the Wu-Zhang system. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. Several examples of linear and nonlinear systems (ODEs and PDEs) are given, including the Wu-Zhang (WZ) system. The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential equations. It also holds great promise for additional real-world applications.