Transmutation Operator Method for Solving Heat Conduction Problem

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.


Introduction
The transmutation operator establishes a connection between the solutions of two problems of mathematical physics,one of the problems is considered as a model one.
Example 1. The Cauchy problem for the heat equation [6] ( ) ( ) 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  Transmutation operator 1 :  In the model problem, the initial condition is chosen in the form In more detail, the inverse In the authors' work [1,2,21], the formula for solution of problem (3) -(4) is transformed to the form 2 2 (7) is a generalization of the Poisson formula of heat conduction theory [6].
In works [3,4,5,7,14,16,18,20] the theoretical foundations of transmutation operator's method are created. Papers [8,[10][11][12][13]15,17,19] are devoted to various applications of the method, including the solving of partial differential equations. The most important of the applications of the developed method is the solution of boundary value problems in areas with axial symmetry [3,11,12]. The transmutation operator method allows problems with axial symmetry to be reduced to boundary value problems in domains with plane symmetry.

Construction of transmutation operator
It will be proved here that the transmutation operator J represents as linear combination of stretched or compressed transforms and reflections.
Proof. Equations (1) The third condition is established similarly. To check the second condition, we calculate the derivatives in the direction of the normal ( ) ( ) Further, the fourth conjugation condition is proved similarly. The conjugation conditions for the functions 12 21 22 , u u u . , Proof. From the first equation of (13) we determine the function ( ) , u x y  in the domains 22 21 12 , , . D D D In the next section, for a better understanding of the construction of the transfmutation operator, we will establish the possibility of its decomposition as a product of simple transfmutation operators 1 J and 2 J .

Factorization of the transmutation operator
The proof follows from Theorem 1.

Extending the Results to the Vector Case
Let a vector-function ( ) ( ) ( ) ( ) ( ) 1 2 , be continuous in, continuously differentiable in R , and continuously differentiable in The parameters  1   2  1  2 , , , a a λ λ are matrices of n n × size for each of which all eigenvalues are different and positive.
The definition of a matrix argument function can be found in the book [9]. We present an unknown definition of the vector function of a matrix argument.
Let us explain the last formula. First, you need to calculate the product ( ) Bf x , and then to apply the definition of a vector-function of a matrix argument. However, the order of operations cannot be changed.
Example 4. If we have matrices Note, if matrices A and B are permute the equality is true.
The vector analogs of transformation operators have the following form: Transmutation operator 1 : ( ) H x − Heaviside step function, and f is defined by equality ( ) ( ) The inverse vector transmutation operator 1 1 J − for the operator 1 J has the form The vector analogue of the initial-boundary value Cauchy problem (3) -(4) for the heat equation is posed as follows: find a solution to the system of differential equations ( ) ( )

Solution of the initial-boundary value problem for the heat equation in an unbounded piecewise-homogeneous plate
We find the solution to the initial-boundary value problem (1) -(3) using Theorem 1. We apply the transmutation operator J from theorem 1 to problem (1) -(3). In the images, we obtain problem (4) -(5), in which the initial state of the temperature field is determined by the formula x η x η x η x η The solution to problem (16) -(17) defines the kernel of the inverse Fourier integral transform 1 1x F − with a point of division on the real axis [21] ( ) ( ) ( ) ( ) 1 1 , .
x x x x ϕ ϕ ϕ ϕ * * * * λ λ (19) The solution to this dual problem (18) - (19) defines the kernel of the direct Fourier integral transform 1x F with a point of division on the real axis ( ) ( ) ( ) ( ) 1 , . x The expressions for the kernels of integral transforms are written out in [21].
The direct 1y F and inverse 1 1y   (9) -(11), we apply the Fourier integral transforms 1x F and 1y F . According to the scheme proposed in [21], we obtain a formula for solving problem (9) Comparing formulas (15) and (20), we conclude the advantage of transmutation operator's method. In formula (15), the solution is given by a double integral, and in formula (20) it is need to use four -times integral. Thus, the transmutation operator's method has all the advantages of the integral transforms method and is more efficient from a computational point of view.