Solutions of Heat-Conduction Problems With Nonseparable Domains

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Abstract
A method is presented for obtaining eigenfunctions of and solutions to the transient heat-conduction equation for a wide class of three-dimensional convex hexahedral domains and two-dimensional convex quadrilateral domains having straight or curved boundaries for which separation of variables cannot be applied. The method is employed to solve for the temperature distribution in a trapezoidal domain, initially at zero temperature, the boundaries of which are subjected to suddenly applied values at the initial instant. The solution is obtained in the form of a series and an examination of successive terms indicates fairly rapid convergence; it is found that the one-term solution yields almost as good values as a four-term solution, which is significant since the former is obtained with little effort. An independent method is utilized for obtaining the steady-state solution, i.e., t → ∞, and it is found that all approximations by the former method are substantially equal to the correct value for this case.

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