ABSTRACT
This chapter discusses one-dimensional steady-state heat conduction problems and observed that formulations of such problems involve ordinary differential equations. Formulations of two- or three-dimensional steady-state and one-dimensional or multi-dimensional unsteady-state heat conduction problems, on the other hand, involve partial differential equations. The chapter describes the solutions of some typical linear heat conduction problems using the method of separation of variables, which was first introduced by d'Alembert, Bernoulli, and Euler in the middle of the 18th century. It also discusses the basic mathematical concepts related to these two methods. The chapter introduces some representative time-dependent problems by Laplace transforms. It develops Fourier transforms, which we implement as another method of solution for linear heat conduction problems. As in the case of many physical problems, in heat conduction problems certain specified conditions must be satisfied by the solution of the relevant form of the heat conduction equation.
