ON THE USE OF VARIATIONAL METHODS FOR SOLVING A CLASS OF PARTIAL DI FERENTIAL EQUATIONS

Abstract

The main objective of this work is the development of the approximate variational techniques for solving the unsteady­ state two dimensional heat diffusion problems. The tradition­ al variational methods can provide solutions to this class of problems of significant accuracy which is adequate to many practical purposes. Two goals are aimed to; the first is the study of the variational calculus while the second is the ap­ plication of a variational method to a heat diffusion problem.

The variational calculus is introduced through the study of its properties, the principles and the fundamental theory of the variational calculus. The major problem of seeking the ex­ tremizing value of a functional is discussed including the ne­ cessary and the sufficient conditions for extremum. The Eul­ er-Lagrange equation is derived and discussed for various types of functionals. This study is ended by the development of the variational formulation and the different classical vari­ ational methods with the application to; a linear and a non­ linear ordinary differential equations, and a steady-state two dimensional heat diffusion problem.

This work is ended by the application of the Galerkin method to the problem of the unsteady-state heat transfer of a flow in a plane channel. The finite-difference method is em­ ployed to solve the problem. The variational solution is com­ pared to the fmite-difference solution and the accuracy of the variational technique is judged.