Eigenfunctions: Expansions into Orthogonality
Partial differential equations are used to model physical phenomena such as heat dissipation, wave propagation, and stress. When the parameters for these events are homogeneous, i.e. the net boundary contribution is zero, and there is no source driving subsequent outcomes, then we use the method of "Separation of Variables" to analytically find solutions. However, in physics, things are not always simple. Inhomogeneous boundaries as well as sources abound in most practical problems. I will demonstrate a technique which utilizes the orthogonal properties of eigen-functions to find analytic solutions to these inhomogeneous PDE's.
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