Transmutation Operator Method for Solving Heat Conduction Problem

Abstract

The transmutation operator method is extended to the case of functions of two variables. The transmutation operator flattens the function, i.e. the transmutation operator replaces a function with discontinuous partial derivatives on the coordinate axes by a continuously differentiable function. The work reveal the properties of the transmutation operator, and prove the commutativity of the transmutation operator and the Laplace operator. It was found that the Cauchy problem for the Laplace equation with internal conjugations in an unbounded domain can be replaced with the model Cauchy problem for the twodimensional Laplace equation. As a result, a new analytical method for solving initial-boundary value problems for a two-dimensional heat equation has been developed. The factorization of the transmutation operator is established as a product of two one-dimensional transmutation operators. The form of the transmutation operator establishing the isomorphism of two mathematical models of heat conduction in unbounded media with different physical characteristics was found and descrfibed.