Matrix Fourier Transforms for Consistent Mathematical Models

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fouriermatrix transformswith discontinuous coefficients are established.We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, andmatrix transforms with piecewise trigonometric kernels.The integral transforms of such kinds are used for problems solving ofmathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.


Introduction
Matrix integral Fourier transforms with sine, cosine, and piecewise trigonometric kernels represent an important branch of mathematical analysis.It is based on the expansion of a function over a set of cosine or sine basis functions.Integral Fourier transforms of such kinds have shown their special applicability in description of consistent mathematical models.To show the versatility of these transforms, we solve the problems of mathematical physics in homogeneous and piecewise homogeneous media.We find analytical solutions of iterated heat conduction equation and solve a problem about stress in the elastic semi-infinite solid.
Given a real function (), which is defined over the positive real line  ≥ 0, for  ≥ 0, which is piecewise continuous and absolutely integrable over [0, ∞), the Fourier  transform of () is defined as subject to the existence of the integral.The inverse Fourier  transform is given by again subject to the existence of the integral used in the definition.The functions () and f(), if they exist, are said to form a Fourier cosine transform pair.Given a real function (), which is defined over the positive real line  ≥ 0, for  ≥ 0, which is piecewise continuous and absolutely integrable over [0, ∞), the Fourier  transform of () is defined as subject to the existence of the integral.The inverse Fourier transform is given by again subject to the existence of the integral used in the definition.The functions () and f(), if they exist, are said to form a Fourier transform pair.Given a real function (), which is defined over the positive real line  ≥ 0, for  ≥ 0, which is piecewise continuous and absolutely integrable over [0, ∞), the Fourier type transform of () is defined as subject to the existence of the integral.

Theorem 1. The unit normalization constant used here provides a definition for the inverse Fourier type transform, given by
again subject to the existence of the integral used in the definition.
In order to define integral Fourier matrix transforms with piecewise trigonometric kernels, we consider Sturm-Liouville matrix problem: where bounded nontrivial unknown matrix function of size  ×  called matrix eigenfunction of Sturm-Liouville problem, () is the matrix-valued function of size  × , and In general, Sturm-Liouville matrix problem does not possess an analytical solution.Therefore, we consider the Sturm-Liouville piecewise approximation as follows.() is piecewise constant; that is, where  =   ,  = 1, . . .,  + 1, are points of discontinuity in  +  , and () is the Heaviside step function.
The elements of matrix eigenfunctions (, ) of a Sturm-Liouville matrix problem are piecewise trigonometric functions.The explicit expression of spectral matrix-valued function (, ) allows for defining direct integral Fourier matrix transform with piecewise trigonometric kernels.The explicit solution of dual Sturm-Liouville matrix problem serves as a kernel for an inverse integral Fourier matrix transform.
Integral transforms arise in a natural way through the principle of linear superposition in constructing integral representations of linear differential equations solutions [13][14][15].The theory of integral Fourier transforms with piecewise trigonometric kernels in a scalar case was studied by Ufljand [16], Najda [17], Procenko and Solov'jov [18], and Lenjuk [19].The matrix version is adapted for the problems solving in piecewise homogeneous medium and has been developed by Yaremko in [10,20].The necessary proofs by method of contour integration were conducted in [11,21].It is clear that this method is effective to obtain the exact solution of boundary value problems for piecewise homogeneous media.

Matrix Fourier Transforms with Piecewise Trigonometric Kernels
The Sturm-Liouville matrix problem [1] is to find the nontrivial solution  bounded on the set  + to a system of an ordinary differential equations with constant matrix coefficients with the boundary conditions at the points  0 and  +1 , and internal boundary conditions at the points  = where Let  be nontrivial solution to boundary value problems ( 52)-( 69) for some .The number  is called an eigenvalue, and the corresponding solution (, ) is called matrix-valued eigenfunction.
We will required invertible conditions for matrices The matrices  2  are positive-defined [4].We denote We define the other  pairs of matrix-value functions (Φ  , Ψ  ),  = 1,  by the induction relations: Introduce the following notations: Lemma 2. The following identity holds true for  = 1, . . ., .
(32) Theorem 4. The spectrum of problems ( 15), (16), and ( 18) is continuous and fills semiaxis (0, ∞).Sturm-Liouville problem is  time singular.Exactly  linearly independent matrix-valued functions correspond to each eigenvalue .It is possible to take  columns of matrix-value functions: That is Theorem 4 follows from Lemmas 2 and 3.
Now we consider the dual matrix Sturm-Liouville problem.We find the nontrivial solution  * of a system of ordinary differential equations with constant matrix coefficients with boundary conditions at the points  =  0 and  =  +1 We write the solution of the boundary value problem in the following form: (39) Theorem 5.The spectrum of problems ( 35), (36), and ( 38) is continuous and fills semiaxis (0, ∞).Sturm-Liouville problem is  time singular.Exactly  linearly independent matrix-valued functions correspond to each eigenvalue .It is possible to take  rows of matrix-value functions: That is, Theorem 5 follows from Lemmas 2 and 3.
The explicit expression of spectral matrix-valued function (, ) and the dual spectral function  * (, ) allow for writing the decomposition theorem on the set  +  .
Theorem 6.Let vector-valued function () be defined over  +  , continuous, absolutely integrable, and has a bounded variation.Then, for any  ∈  +  , the decomposition formula This theorem can be proved by method of contour integration [17].
We define the direct and inverse matrix integral Fourier transforms on the real semiline with piecewise trigonometric kernels according to Theorem 6.
The direct transform is and the inverse transform is when Now we will get the result of the basic identity of matrix integral transforms with piecewise trigonometric kernels for differential operator: and internal boundary conditions at the points  =   , then the basic identity This theorem can be proved by method of integration by parts.

Special Types of Matrix Integral Fourier Transforms Theorem 8. The matrix-valued Sturm-Liouville problem with Dirichlet boundary condition
provides the direct and inverse matrix integral  transforms on the real semiline: Proof.Performing calculations in formulas (33) and (41), we get provides the direct and inverse matrix integral  transforms on the real semiline: Theorem 10.The matrix-valued Sturm-Liouville problem with Robin boundary condition where  is the square matrix with negative eigenvalues, provides the direct and inverse matrix Fourier type transforms on the real semiline: Proof.Substitute into ( 33) and (41); then,

Theorem 11. The matrix-valued Sturm-Liouville problem with Dirichlet boundary condition on the composite semiline 𝐼
provides the direct and inverse matrix sine type integral transforms on the composite real semiline: where Proof.Performing calculations in formulas (33) and (41), we get Then, matrix eigenfunctions of Sturm-Liouville problem have the following form: And dual matrix eigenfunctions of Sturm-Liouville problem have the following form: Now we can use the matrix sine type integral transforms on the composite real semiline (43) and (44) to describe analytically the consistent mathematical models.

Analytical Solution of Iterated Heat Conduction Equation
In this section, we can solve a mixed boundary value problem for iterated heat conduction equation [22].Let be a solution of system of differential equations with initial conditions with boundary conditions At the beginning, we will solve an auxiliary vector mixed boundary value problem.Let be a solution to the system of the differential equations with initial conditions with boundary conditions at the point  = 0 In the case of plane, the strain vector of displacement   has components   , V  , 0. Introduce Airy stress function [10] as a solution to system of differential equations with boundary conditions and internal boundary conditions at the points  = where   is the normal stress and   is the shearing stresses.
is the solution to system of differential equations with boundary conditions and internal boundary conditions at the points  = At the beginning, we will solve an auxiliary vector mixed boundary value problem.Let be a solution to the system of differential equations with boundary conditions and internal boundary conditions at the points  =   ( 1 0  Calculating the components   and V  of strain vector of displacement on the basis of [10], we get  (110)

Conclusion
Usage of the integral Fourier matrix transforms with piecewise trigonometric kernels method allows us to solve internal boundary conditions problems.Internal boundary conditions problems arise in mathematical modeling of heat conduction and stress produced in the piecewise homogeneous media.

Lemma 15 .
The solution of problems (101)-(104) in the Fourier images takes the form  −1 + is constructed in accordance with (44).