We study the Cauchy problem for the linear differential equation ∂u/∂t = Fu in the module of formal generalized functions over a commutative ring, where F is a differential operator of finite or infinite order. We prove the well-posedness of the problem, find its fundamental solution, and obtain a representation of a unique solution to the Cauchy problem in the form of the convolution of the fundamental solution and initial condition. These results are specified for the heat equation, the simplest transport equation and the one-dimensional wave equation.
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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 27-41.
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Gefter, S.L., Piven’, A.L. Linear Partial Differential Equations in Module of Formal Generalized Functions over Commutative Ring. J Math Sci 257, 579–596 (2021). https://doi.org/10.1007/s10958-021-05505-0
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DOI: https://doi.org/10.1007/s10958-021-05505-0