The Estimation For Recent Trends In Special Partial Differential Equations and Its Applications |

ABSTRACT:

In the last ten years, there has been significantimprovement and growth in tools that aid the development of finite element methodsfor solving partial differential equations. These tools assist the user intransforming a weak form of a differential equation into a computable solution.Despite these advancements, solving a differential equation remainschallenging. Not only are there many possible weak forms for a particularproblem, but the most accurate or most efficient form depends on the problem’sstructure. Requiring a user to generate a weak form by hand creates asignificant hurdle for someone who understands a model, but does not know howto solve it. In this article a symmetry group of scalingtransformations is determined for a partial differential equation of fractionalorder α, containing amongparticular cases the diffusion equation, the wave equation, and the fractionaldiffusion-wave equation. The conventional differential quadrature (DQ) method islimited in its application to regular regions by using functional values alonga mesh line to approximate derivatives. In this work, we extend the idea of DQmethod to a general case. In other words, any spatial derivative isapproximated by a linear weighted sum of all the functional values in the wholephysical domain. In the last ten years, there has been significant improvementand growth in tools that aid the development of finite element methods forsolving partial differential equations. These tools assist the user intransforming a weak form of a differential equation into a computable solution.