Keywords: fourier matrix transforms, integral transform, heat conduction equation, wave equation, Poisson equation
International Journal of Partial Differential Equations and Applications, 2014 2 (5),
pp 9195.
DOI: 10.12691/ijpdea252
Received October 20, 2014; Revised November 12, 2014; Accepted November 18, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
Matrix 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 transforms arise in a natural way through the principle of linear superposition in constructing integral representations of solutions to linear differential equations. The theory of integral Fourier transforms with piecewise trigonometric kernels in a scalar case was studied by Ufljand J.S. ^{[1, 2]}, Lenjuk M. P ^{[3]}, Najda L.S. ^{[4]}, Protsenko V. S ^{[5]}. The matrix version is adapted for the solution to the problems in piecewise homogeneous medium and have been developed by O.Yaremko in ^{[6, 8, 9]}. The necessary proofs by method of contour integration were conducted in ^{[6]} and ^{[9]}. It is clear this method is effective to obtain the exact solution of boundaryvalue problems for piecewise homogeneous media ^{[7, 10, 11, 13]}. Special types of matrix Fourier transforms on real axis and semiaxis and their applications are analyzed in this article.
Fourier transforms with sine, cosine and piecewise trigonometric kernels have shown their special applicability in description of consistent mathematical models. To show the versatility of these transforms constructed by authors we solve the vector problems of mathematical physics in homogeneous and piecewise homogeneous media. We find analytical solutions on real axis and real semi axis of classical mathematical models:vector heat conduction equation,vector wave equation and vector Poisson equation.
Now we consider classical SturmLiouville problems of variety types in boundary conditions. Further we construct their matrix analogues in section 2,3.
The SturmLiouville problem with Dirichlet boundary condition
provides the kernels for direct and inverse integral  transforms on the real semiaxis:
The SturmLiouville problem with Neumann boundary condition
provides the kernels for direct and inverse integral  transforms on the real semiaxis:
The SturmLiouville problem with Robin boundary condition
where  negative, provides the kernels for direct and inverse Fourier type transforms on the real semiaxis:
 (1) 
 (2) 
Theorem 1. If in (1), (2) then the inverse Fourier type transform has the form
subject to the existence of the integral used in the definition.
Proof. Let function takes the form
Then
Due to the inverse sin transform, we get
Note. If is a vector function and is a square negative definite matrix then theorem 1 is true.
2. Matrix Fourier Transforms on Real Axis
In linear algebra, a symmetric real matrix is said to be positive definite if is positive for every nonzero column vector of real numbers. Here denotes the transpose of . Let be an eigendecomposition of , where is a unitary real matrix whose rows comprise an orthonormal basis of eigenvectors of , and is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. Let where
Definition 1. Let be a vectorfunction
and be the matrix give an account of above, then vectorfunction of matrix is defined by the following equality
Theorem 2. The matrix value SturmLiouville problem
provides the direct and inverse matrix Fourier transforms on the real axis:
Theorem 3. The matrix value SturmLiouville problem on the composite axis
provides the direct and inverse matrix integral transforms on the composite real axis
where
All necessary proofs were conducted in ^{[6, 9, 15]} by method of contour integration.
3. Matrix Fourier Transforms on Real Semiaxis
Theorem 4. The matrix value SturmLiouville problem with Dirichlet boundary condition
provides the direct and inverse matrix integral  transforms on the real semiaxis:
Theorem 5. The matrix value SturmLiouville problem with Neumann boundary condition
provides the direct and inverse matrix integral  transforms on the real semiaxis:
The kernels of integral transforms in theorems 4,5 may be calculated by formulasfrom ^{[15]}.
Theorem 6. The matrix value SturmLiouville problem with Robin boundary condition
where  square matrix with negative eigenvalues, provides the direct and inverse matrix Fourier type transforms on the real semiaxis:
Proof. Let
then
Applyingtheorem 1, замечание 1исчитаяприэтом whence we may write
Using the definition offunction , we obtain
4. Vector Wave Equation
In this section we can solve Cauchy problem for vector wave equation
The solution of this Cauchy problem has the form
Simplify this formula we get
So vector version of d'Alembert formula ^{[14]} is obtained.
5. Vector Heat Equation
In this section we can solve Cauchy problem for vector heat equation
The solution of Cauchy problem has the form
So vector version of Poisson's formula is obtained.
We consider Cauchy problem for vector heat equation on real composite axis
with internal boundary conditions at the point
The solution of this problem has the form
Then using theorem 3 we obtain
where
If we change the order of integration we obtain vector version of Poisson's formula ^{[14]}.
6. Vector Dirichlet Problem
In this section we can solve Dirichlet problem for vector Laplace equation on real axis and real composite axis. Dirichlet problem on real axis is given by
It solution is
Dirichlet problem on real composite axis for vector Laplace equation is given by
with boundary conditions
with internal boundary conditions at the point
The solution of this problem be as follows
If we change order of integration we obtain vector version of Poisson's formula ^{[14]}
where
is Poisson kernel ^{[14]}.
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