Solving the heat equation in piecewise-homogeneous anisotropic media using the multidimensional Fourier transforms

TL;DR

This paper develops multidimensional integral transforms with explicit kernels to solve the heat equation in media with discontinuous, anisotropic coefficients across hyperplanes, advancing analytical methods for complex heterogeneous materials.

Contribution

It introduces explicit formulas for integral transform kernels and proves the fundamental identity, enabling solutions in piecewise-homogeneous anisotropic media with discontinuities.

Findings

Explicit kernel formulas for integral transforms with discontinuous coefficients

Proof of the fundamental identity of the integral transform

Development of a technique for solving heat equations in complex media

Abstract

Multidimensional integral transformations with non-separated variables for problems with discontinuous coefficients are constructed in this work. The coefficient discontinuities focused on the of parallel hyperplanes. In this work explicit formulas for the kernels in the case of ideal coupling conditions are obtained; the basic identity of the integral transform is proved; technique of integral transforms is developed