Transmutation Operator Method for Solving Heat Conduction Problem

Oleg Yaremko,Natalia Yaremko

doi:10.1051/epjconf/202124801019

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.

Highlights

The transmutation operator establishes a connection= uu= 22t (0,ax22)u2 xx, t f2 > (x ), < < x x ∞ ∞ (3)between the solutions of two problems of mathematical physics,one of the problems is considered as a model with internal conjugation conditions one
Between the solutions of two problems of mathematical physics,one of the problems is considered as a model with internal conjugation conditions one
Let us set the goal of obtaining the Poisson formula for solving the heat equation in an unbounded four-layer plate

Summary

Introduction

Between the solutions of two problems of mathematical physics,one of the problems is considered as a model with internal conjugation conditions one. An operator J : u → u that establishes an isomorphism of the solution space for problems (1) and (2) is called a transmutation operator. In this example, the transmutation operator is defined by the formula. Consider the initial-boundary value Cauchy problem for the heat equation ν = 1 λ2 a1 . Let u (t, x) be the solution of the initialboundary value problem (3) - (4) with parameters a 1, a 2 , λ 1, λ 2 and initial condition. U (t, x) is the solution to initial-boundary value problem (3)-(4) with parameters a1, a2 , λ1, λ2 and initial condition.

Statement of the problem

Construction of transmutation operator

Discussion

Conclusion