Different numerical methods offer for solving wave equation with diverse boundary conditions. In these methods given partial (Differential Equation: DE) change to a normal DE system using equation separating in terms of time variable method. PADE APPROXIMATION is used for appreciative calculation of exponential matrix function which is created in response of normal DE. Furthermore, its error and stability are investigated by separating method, parallel algorithm is proposed for solving normal Differential Equation and numbers of function in serial and parallel methods are compared.

The combined effects of nanoparticles and magnetic field on the nonlinear Jeffery-Hamel flow are analyzed in the present study. The basic governing equations are solved into series solution using a semi-numerical analytical technique called Hermite- PADE APPROXIMATION. The velocity profiles are presented in divergent channel for various values of nanoparticles solid volume fraction, Hartmann number, Reynolds number and channel angle. The dominating singularity behavior of the problem is analysed numerically and graphically. The critical relationship between the parameters is studied to observe the instability of the problem for nanofluid.

The Ising model is one of the simplest models describing the interacting particles. In this work, we calculate the high temperature series expansions of zero field susceptibility of ising model with ferromagnetic, antiferromagnetic and one antiferromagnetic interactions on two dimensional kagome lattice. Using the PADE´ APPROXIMATION, we calculate the susceptibility of critical exponent of ferromagnetic ising model  g » 1.75, which is consistent with universality hypothesis. However, antiferromagnetic and one antiferromagnetic interaction ising model doesn’t show any transition at finite temperature because of the effect of magnetic frustration.

In this paper, analytic solutions of two-dimensional coupled Burgers’ equations are obtained by the Homotopy analysis and the Homotopy PADE methods. The obtained APPROXIMATION by using Homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The APPROXIMATION solutions by [m,m] Homotopy PADE technique is often independent of auxiliary parameter and this technique accelerate the convergence of the related series.

In this paper, we are giving analytic approximate solutions to nonlinear PDEs using the Homotopy Analysis Method (HAM) and Homotopy PADE Method (HPADEM). The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence regions of solution series. It is illustrated that HPADEM accelerates the convergence of the related series. The results reveal these methods are remarkably effective.