Analytical solutions for diffusive finite reservoir problems using a modified orthogonal expansion method

In this paper, we consider an eigenfunction expansion solution method useful for a family of 1-d, unsteady diffusion dominated transport problems (Mass, e.g., effluent leakage from a landfill; Momentum, e.g., deceleration of shaft in a journal bearing, and energy; e.g., unsteady thermal loading of the skin of a re-entry vehicle) characterized by movement from a bounded, and therefore, time dependent, reservoir to an infinite reservoir. It is demonstrated that simple orthogonality is not applicable to this type of problem. To overcome this limitation, an extended eigenfunction expansion method is developed by modifying the weighting function within the classical orthogonality definition. Using this method, an analytical series solution is obtained. This series solution is also obtained using Laplace transform methods and analytical inversion. Comparison with numerical and approximate methods is good. The analytical solution developed here provides a convenient and physically insightful solution form useful in its own right and as a test problem for numerical implementations.