Abstract
We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical 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.
1. 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 , for , which is piecewise continuous and absolutely integrable over , the Fourier transform of is defined assubject to the existence of the integral. The inverse Fourier transform is given byagain subject to the existence of the integral used in the definition. The functions and , if they exist, are said to form a Fourier cosine transform pair.
Given a real function , which is defined over the positive real line , for , which is piecewise continuous and absolutely integrable over , the Fourier transform of is defined assubject to the existence of the integral. The inverse Fourier transform is given byagain subject to the existence of the integral used in the definition. The functions and , if they exist, are said to form a Fourier transform pair.
Given a real function , which is defined over the positive real line , for , which is piecewise continuous and absolutely integrable over , 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 byagain subject to the existence of the integral used in the definition.
Proof. Let function take the following form:Then,Due to the inverse Fourier transform, we getThe general theory of linear integral transform with some of its applications gave an account of [1–12].
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 , andIn 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 , 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 [13–15]. 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.
2. 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 coefficientswith the boundary conditions at the points and ,and internal boundary conditions at the points where are matrices of size :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 conditionsfor matricesThe matrices are positive-defined [4]. We denoteWe define the other pairs of matrix-value functions by the induction relations:Introduce the following notations:
Lemma 2. The following identityholds true for .
Proof. Using formula [20]and identitywe can conclude
Lemma 3. If the following inequalityholds true, then
Proof. Matrices and are nonsingular due to the following identity:
Theorem 4. The spectrum of problems (15), (16), and (18) is continuous and fills semiaxis . 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 coefficientswith boundary conditions at the points and and internal boundary conditions at the points ,We write the solution of the boundary value problem in the following form:
Theorem 5. The spectrum of problems (35), (36), and (38) is continuous and fills semiaxis . 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 formulaholds true.
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 isand the inverse transform iswhen
Now we will get the result of the basic identity of matrix integral transforms with piecewise trigonometric kernels for differential operator:
Theorem 7. If vector-valued functionis three times continuously differentiable over set , has the limit values of its third-order derivatives at the points , and satisfies boundary condition on infinityand internal boundary conditions at the points , then the basic identityholds true.
This theorem can be proved by method of integration by parts.
3. Special Types of Matrix Integral Fourier Transforms
Theorem 8. The matrix-valued Sturm-Liouville problem with Dirichlet boundary conditionprovides the direct and inverse matrix integral transforms on the real semiline:
Proof. Performing calculations in formulas (33) and (41), we get
Theorem 9. The matrix-valued Sturm-Liouville problem with Neumann boundary condition provides the direct and inverse matrix integral transforms on the real semiline:
Theorem 10. The matrix-valued Sturm-Liouville problem with Robin boundary conditionwhere 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 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 getThen, 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.
4. Analytical Solution of Iterated Heat Conduction Equation
In this section, we can solve a mixed boundary value problem for iterated heat conduction equation [22].
Letbe a solution of system of differential equationswith initial conditionswith boundary conditionsand 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 the differential equationswith initial conditionswith boundary conditions at the point and internal boundary conditions at the points
Lemma 12. The solution of problems (72)–(76) has the following form:where
Proof. This lemma can be proved by Fourier integral transform method with piecewise trigonometric kernels (43)-(44). Let the vector-functionbe the Fourier transform. Then, from Theorem 7, vector-function will be a solution of the Cauchy problem: The solution has the following form:To complete the proof, we apply inverse Fourier transform :
Theorem 13. Let be a solution of vector problems (72)–(76); then,is a solution to mixed boundary value problems (67)–(70) for iterated heat conduction equation (67).
Proof. In accordance with (72), the function is a solution of iterated heat conduction equation (67):Due to (76), we get the initial condition: On the basis of (74), the Robin boundary conditions have the following form: the first condition is as follows:and the second condition is as follows: if , then It follows from (93)-(94) thatThen, the internal boundary conditions at the points hold true.
Corollary 14. The solution of problems (67)–(70) has the following form:
5. Stress Produced in the Elastic Semi-Infinite Solid by Pressure
Let us consider a problem about distribution of tension in an -layer elastic semi-infinite solid:
In the case of plane, the strain vector of displacement has components . Introduce Airy stress function [10]as a solution to system of differential equationswith boundary conditionsand internal boundary conditions at the points where is the normal stress and is the shearing stresses. Fourier transform of Airy stress function with respect to is the solution to system of differential equationswith boundary conditionsand 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 equationswith boundary conditionsand internal boundary conditions at the points
Lemma 15. The solution of problems (101)–(104) in the Fourier images takes the formand in the Fourier originals has the form
Inverse Fourier transform is constructed in accordance with (44).
Theorem 16. Let be a solution of vector problems (101)–(104); then,is the solution of scalar problems (75)-(76).
Proof. In accordance with (101), the function p is a solution of iterated Laplace equation (93):Calculating the components and of strain vector of displacement on the basis of [10], we getAs a result, the internal boundary conditions (103)-(104) hold.
Corollary 17. The solution of problems (93)–(95) has the form
6. 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.
Competing Interests
The authors declare that they have no competing interests.