Nonhomogeneous Problem and Eigenfunction Expansion

In the previous section, the method of separation of variables was applied to some problems with special nonhomogeneous terms that could be recasted as homogeneous by using a suitable substitution. In this section, a method will be outlined that is applicable to those nonhomogeneous problems for which no simple substitution can be made to remove the nonhomogeneity. This method is called eigenfunction expansion [3,4,6]. 

Consider the flow of heat in a rod of length L that is uniformly constructed (Equation 6.11, Section 6.2). Further, the rod has temperature-independent heat sources distributed in some prescribed way throughout and is time-depen-dent. In addition, the temperatiue at the ends is allowed to be time-dependent. Then, for a prescribed initial temperature distribution, the following model is appropriate  

Equation 6.60 to Equation 6.62 describe a nonhomogeneous PDE with nonhomogeneous boundary conditions. The associated homogeneous model is given by  

However, term-by-term time derivatives are valid, such that 

Further, recall that — = — with t held constant. Then the formula 

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