In this paper we applied the tanh method for analytic study of the nonlinear equations of partial differential equations(PDEs).The proposed method gives more general exact traveling wave solutions without much extra effort. Three applications from literature of nonlinear equation of PDEs were solved by the method. The calculations demonstrate the effectiveness and convenience of the method for nonlinear sub system of PDEs.

The present paper concerns a theoretical study of heat transfer of the laminar two dimensional flows of various nanofluids taking into account the dissipation due to viscous term past a 2-D flat plate had a different temperatures. The steady incompressible flow equations were used and transformed to a nonlinear Ordinary Differential Equation (ODE) using a similarity variable. These equations were solved numerically using implicit finite difference method in which the partial derivatives were replaced by appropriate central differences patterns and using Newton's method to linearize the resulting algebraic equations. Finally, the block-tridiagonal-elimination technique was used to solve that linear system. Three types of nanoparticles namely, Cu-water, Al2O3-water, and TiO2-water in the base flow of water were considered. The symbolic software Mathematica was used in the present study. Different types of nanoparticles, different values of, nanoparticle volume fraction, Eckart and Prandtl number were tested and analyzed at different wall temperature. The effect of these parameters on the flow behaviour, the local skin friction coefficient, Nusselt number, the velocity and the temperature profiles were presented and investigated. It is concluded that these parameters affect the fluid flow behaviour and heat transfer parameters especially nanoparticle concentration. The presence of nanoparticles showed an enhancement in the heat transfer rate moreover its type has a significant effect on heat transfer enhancement.

A finite element solution of the modified Korteweg-de Vries (MKdV) equation, based on Galerkin’s method using cubic splines as element shape functions, is set up. A linear stability analysis shows the scheme is unconditionally stable. Numerical tests for single, two and three solitons are used to assess the performance of the proposed scheme. The four invariants of motion are evaluated to determine the conservation properties of the algorithm.