With iterated error functions we have already greatly expanded our vocabulary of solutions of the diffusion equation. The question is, however, whether or not any useful further expansion should be expected. The study of physical symmetries provides a partial answer, and that study is called similarity. In essence, we shall find that the linear diffusion equation in one dimension has two types of solutions: superpositions either of iterated error functions or of trigonometric functions, otherwise known as Fourier series. The strength of similarity methods lies in their applicability to nonlinear problems as well. This chapter continues with more physical examples, including exact solutions for the kinetics of certain first-order phase transformations and of diffusion problems where the diffusivity depends on the field. In contrast, Chaps. 7 and 8 describe deductive methods for the solution of diffusion problems.
KeywordsDiffusion Equation Similarity Solution Diffusion Problem Nonlinear Diffusion Stefan Problem
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