Initial-Boundary Value Problem for an Inhomogeneous Heat Equation with Piecewise Constant Delay

Abstract In this paper, an initial-boundary value problem for an inhomogeneous heat equation with a piecewise constant argument and Dirichlet boundary conditions is considered. The Fourier method is used to investigate the problem. By expanding the solution in terms of eigenfunctions, the initial-boundary value problem is reduced to the Cauchy problem for an ordinary differential equation with respect to the expansion coefficients with a piecewise continuous argument. The existence and uniqueness of the solution to this problem are proved. As a result, it is shown that the original problem has a unique solution, which is constructed in explicit form.

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