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Abstract

This paper aims to provide an exact solution in the Laplace domain and related analytic approximations in short time limits for the class of boundary value problems of the one-dimensional linear parabolic equation with constant coefficients. The problem’s most general form involves a parameterized equation on a bounded interval, with unified specification of the three classical types of boundary conditions: Dirichlet, Neumann, and Robin. Under certain integrability assumptions, we have proven that a unique solution exists in the Laplace domain. This operational solution can be obtained in a closed form by using classical integral transforms. Four distinct cases have been identified based on the operational solution. Innovative formulas have been derived from these cases, which provide precise approximations within short timescales. These time-domain expressions are particularly useful for understanding the behavior of the solution at the boundaries. The formulas consist of elementary functions obtained from asymptotic expansions, and the estimation error can be minimized to the desired order of magnitude. The analytical approximations in short time limits can open up new perspectives and applications. Improved numerical efficiency in simulations of reaction-diffusion problems and of one-dimensional Stefan models are envisaged.

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