Green’s function of heat equation for heterogeneous media in 3-D

Abstract

The purpose of the present paper is to study the structure of Green’s function for heat equation in several spatial dimensions and with rough heat conductivity coefficient. We take the heat conductivity coefficient to be of bounded variation in the x direction and study the dispersion in the (y, z) direction. The goal is to understand the coupling of dissipation across rough heat conductivity and the multi-dimensional dispersion in the Green’s function $\mathsf{H}(\overrightarrow{\mathit{x}},t;{\overrightarrow{\mathit{x}}}_{*}),\overrightarrow{\mathit{x}}=(x,y,z).$ A series of exponential functions of path integral with coefficients over a field of complex analytic functions around imaginary axis are formulated in the Laplace and Fourier transforms variables. The Green’s function in the transformed variables is written as the sum of these integrals over random paths. The integral over a random path is rearranged through the reflection property over a variation of heat conductivity coefficient and become a simple form in terms of path phase and amplitude. The complex analytic and combinatorics method is then used to yield a precise pointwise structure of the Green’s function in the physical domain $(\overrightarrow{\mathit{x}},t).$

Keywords:

• Green’s function of heat equation
• heterogeneous media
• Fourier-Laplace transform
• bounded variation

Additional information

Funding

The research of the first author is supported in part by NSTC Grant 111-2115-M-008-007-MY2. The second author by MOST 106-2115-M-001-011- and Simons Foundation Grant, and third author by MOST 110-2115-M-001-017-MY3. MOST Grant (Taiwan).