Mathematical prerequisites
• Orthogonal functions; the Sturm–Liouville equation.
• Fourier expansion of a function and evaluation of the Fourier coefficients.
• The Laplace transform and inverse Laplace transforms.
• Complex representation of a sine or cosine function; manipulations of complex variables.
In Chapters 8 and 10 we examined a number of steady-state problems in heat and mass transfer. In this chapter we examine unsteady-state problems. Owing to the similar mathematical structure heat and mass transfer problems can be treated in a similar manner. The concentration or temperature is now a function of both time and the spatial coordinates, and therefore the governing equations are now partial differential equations. This adds additional mathematical complexity to the solution. The most general problem in 3D and time and with time-varying boundary conditions needs a numerical solution. However, there are many simpler but nevertheless important problems where analytic solutions are possible and useful. This chapter introduces and solves such problems and illustrates a number of important techniques to obtain the solutions.
For linear partial differential equations, analytic solutions based on the method of separation of variables are commonly used. It can be applied to problems where the differential equations and the boundary conditions are homogeneous (as will be explained in the text) and applies to problems posed in a finite spatial domain.
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