On the Simulation of Differential Transform Method using Padeʹ Approximation for Studying Some Mathematical Models concerned with the Heat Transfer and Stability of Fluids

Abstract

Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many analytical methods have been developed to determine solutions with a given degree of accuracy. One of the effective semi-analytical methods is named by the differential transform method (DTM), which is a semi-analytical method for solving integral equations, ordinary and partial differential equations. The method provides the solution in terms of convergent series with easily computable components. In this thesis, we use the generalized methods of DTM by new techniques named by Pade'-approximation differential transform method (Pade'-DTM) and the Multi-step differential transform method (Ms-DTM) to solve the highly nonlinear systems of fluid mechanics, and especially peristaltic flow problems