Our objective in this chapter is to discuss the basic solution techniques associated with the heat equation, wave equation, and Laplace's equation. Each of these three partial differential equations is a simple representative of a certain class of equations, and hence, provides much insight into the general behavior of other equations in the class. Integral transform techniques are utilized for problems in which the spatial domain becomes unbounded.
In general, partial differential equations (PDEs) are prominent in those physical and geometrical problems involving functions that depend on more than one independent variable. These variables may involve time and/or one or more spatial coordinates. Our treatment of such equations is intentionally brief, however, focusing on only the basic equations of heat conduction, wave phenomena, and potential theory. These equations are simple representatives of three important classes of PDEs that include most of the equations of mathematical physics.
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