Variational Solutions of Transient Heat Conduction Through Bodies of Finite Length

Abstract

Publisher Summary The chapter presents the variational principle with vanishing parameter in the study of linear and nonlinear transient heat conduction through bodies of finite length. The two most popular direct methods, partial integration and the Rayleigh-Ritz direct method, are the principal topics of the chapter. The chapter describes how the two direct methods usually employed in variational calculations are tailored to the variational principle with vanishing parameter. The chapter presents approximate solutions for certain particular linear and nonlinear problems that illustrate direct methods based on the variational principle with vanishing parameter. The chapter discusses the selection of a trial solution for transient linear heat-conduction problems with asymmetric boundary condition. In this framework, a method of constructing orthogonal functions of spatial coordinates is presented. The method is illustrated by a linear and a nonlinear example. The chapter demonstrates that the canonical form of heat conduction can be generated by means of the variational principle with vanishing parameter and Lagrangian function. The chapter presents an application of this variational formulation by means of a simple example that reflects the ordinary details in obtaining variational solutions.