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Matrix Fourier Transforms and Application

O. Yaremko, O. Nikitina, E. Zuravleva*

Oleg Yaremko,Olga Nikitina, Ekaterina Zuravleva Penza State University, Penza, Russia

Abstract: In this work, we introduce in an explicit form a special types of matrix Fourier transforms on real axis and real semi- axis: matrix cos- transforms, matrix sin- transforms and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for solving analytically the problems of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of vector heat conduction equation,vector wave equation and vector Poisson equation is obtained.

Keywords: fourier matrix transforms, integral transform, heat conduction equation, wave equation, Poisson equation

Received October 20, 2014; Revised November 12, 2014; Accepted November 18, 2014

Citation: O. Yaremko, O. Nikitina, E. Zuravleva, Matrix Fourier Transforms and Application. International Journal of Partial Differential Equations and Applications, 2 (5). 91-95. January 2014. doi:10.12691/ijpdea-2-5-2

1. Introduction

Matrix Fourier transforms with sine, cosine and piece-wise 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 boundary-value problems for piecewise homogeneous media [7,10,11,13]. Special types of matrix Fourier transforms on real axis and semi-axis and their applications are analyzed in this article.

Fourier transforms with sine, cosine and piece-wise 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 Sturm-Liouville problems of variety types in boundary conditions. Further we construct their matrix analogues in section 2,3.

The Sturm-Liouville problem with Dirichlet boundary condition

provides the kernels for direct and inverse integral - transforms on the real semi-axis:

The Sturm-Liouville problem with Neumann boundary condition

provides the kernels for direct and inverse integral - transforms on the real semi-axis:

The Sturm-Liouville problem with Robin boundary condition

where - negative, provides the kernels for direct and inverse Fourier- type transforms on the real semi-axis:

[picture] | (1)

[picture] | (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 non-zero 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 vector-function

and be the matrix give an account of above, then vector-function of matrix is defined by the following equality

Theorem 2. The matrix -value Sturm-Liouville problem

provides the direct and inverse matrix Fourier transforms on the real axis:

Theorem 3. The matrix -value Sturm-Liouville 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 Semi-axis

Theorem 4. The matrix value Sturm-Liouville problem with Dirichlet boundary condition

provides the direct and inverse matrix integral - transforms on the real semi-axis:

Theorem 5. The matrix value Sturm-Liouville problem with Neumann boundary condition

provides the direct and inverse matrix integral - transforms on the real semi-axis:

The kernels of integral transforms in theorems 4,5 may be calculated by formulasfrom [15].

Theorem 6. The matrix -value Sturm-Liouville problem with Robin boundary condition

where - square matrix with negative eigenvalues, provides the direct and inverse matrix Fourier type transforms on the real semi-axis:

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].

References

[1]   | Uflyand I. S. (1967) Integral transforms in the problem of the theory of elasticity. Leningrad. Science, p. 402
[2]   | Uflyand I. S. (1967) On some new integral transformations and their applications to problems of mathematical physics. Problems of mathematical physics. Leningrad, p. 93-106
[3]   | M.P.Lenjuk, Integral'noe preobrazovanie Fur'e na kusochno-odnorodnoj poluprjamoj, Izv. vuzov. Matematika, Volume 4,1989, p. 14-18.
[4]   | L.S.Najda, Gibridnye integral'nye preobrazovanija tipa Hankelja-Lezhandra, Mat. metody analiza dinam. sistem, Har'kov, Volume 8, 1984, p. 132-135.
[5]   | V.S.Procenko, A.I.Solov'jov, Gibridnye integral'nye preobrazovanija i ih prilozhenija v teorii uprugosti neodnorodnyh sred, Prikladnaja mehanika, Issue 1, Volume 13, 1982, p. 62-67.
[6]   | E. Mogileva, O. Yaremko, Hermite functions with discontinuous coefficients for the solution of fractal diffusion retrospective problems, International journal of applied mathematics and informatics,Issue 3, Volume 7, 2013, p. 78-86.
[7]   | F.M. Mors, G. Fishbah, Methods of theoretical physics, New York, McGraw-Hill, 1953. Part I, 998 pp.
[8]   | O.E. Yaremko, Matrix integral Fourier transforms for problems with discontinuous coefficients and transformation operators, Reports of Academy of Sciences, Volume. 417, Issue 3, 2007, p. 323-325.
[9]   | Bavrin I.I., Matrosov V.L., Jaremko O. E. (2006) Operators of transformation in the analysis, mathematical physics and Pattern recognition. Moscow, Prometheus, p 292.
[10]   | Brejsuell R., (1990) Hartley transform, Moscow, World, p 584.
[11]   | Vladimirov V. S., Zharinov V.V., (2004) The equations of mathematical physics, Moscow, Phys mat lit, p 400.
[12]   | Gantmaxer F.R., (2010) Theory matrix. Moscow, Phys. Mat. Lit., p 560.
[13]   | Sneddon I., (1955) Fourier Transform, Moscow.
[14]   | Physical encyclopedia. The editor-in-chief A. M. Prokhorov, D.M. Alekseev, Moscow, (1988-1998).
[15]   | Jaremko O. E., (2007) Matrix integral Fourier transform for problems with discontinuous coefficients and conversion operators. Proceedings of the USSR Academy of Sciences. p. 323-325.