Our ability to solve certain differential equations increases with our lexicon of available functions. Exponentials, for example, are basic solutions of ordinary linear differential equations with constant coefficients. Likewise, to solve time-dependent problems in diffusion theory requires the introduction of certain “special functions” that are closely related to gaussians.1–3 In this chapter we follow a constructive path that issues from our earlier experience with random walks, through new physical examples, and finally to the family of error functions. We then show, in the next chapter, that these error functions are, in fact, part of a grander scheme—similarity—that also includes the classical method of separation of variables. This chapter ends with a consideration of problems in several space dimensions that can be expressed in terms of products of 1-d problems.
KeywordsAsymptotic Expansion Diffusion Equation Error Function Gamma Function Stefan Problem
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