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Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

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Abstract

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.

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Authors and Affiliations

  1. Aerospace and Mechanical Engineering Dept., University of Arizona, 85721, Tucson, Arizona, U.S.A.

    G. W. Swan

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  1. G. W. Swan
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Swan, G.W. Exact fundamental solutions of linear parabolic equations with spatially varying coefficients. Bltn Mathcal Biology 39, 435–451 (1977). https://doi.org/10.1007/BF02462922

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  • DOI: https://doi.org/10.1007/BF02462922

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