# Matrix Fourier Transforms for Consistent Mathematical Models

Authors: Oleg Yaremko; Natalia Yaremko
Journal: Chinese Journal of Mathematics
 (2016)

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## 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.


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## Body

## 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 functionf(x), which is defined over the positive real line x≥0, for λ≥0, which is piecewise continuous and absolutely integrable over [0,∞), the Fourier cosine transform of f(x) is defined as(1)Ffλ=f~λ=∫0∞cos⁡λxfxdxsubject to the existence of the integral. The inverse Fourier cosine transform is given by(2)F-1f~λ=fx=2π∫0∞cos⁡λxf~λdλagain subject to the existence of the integral used in the definition. The functions f(x) and f~(λ), if they exist, are said to form a Fourier cosine transform pair.Given a real functionf(x), which is defined over the positive real line x≥0, for x≥0, which is piecewise continuous and absolutely integrable over [0,∞), the Fourier sine transform of f(x) is defined as(3)Ffλ=f~λ=∫0∞sin⁡λxfxdxsubject to the existence of the integral. The inverse Fourier transform is given by(4)F-1f~λ=fx=2π∫0∞sin⁡λxf~λdλagain subject to the existence of the integral used in the definition. The functions f(x) and f~(λ), if they exist, are said to form a Fourier transform pair.Given a real functionf(x), which is defined over the positive real line x≥0, for x≥0, which is piecewise continuous and absolutely integrable over [0,∞), the Fourier type transform of f(x) is defined as (5)Ffλ=f~λ=∫0∞hsin⁡λx-λcos⁡λxfxdxsubject 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(6)F-1f~λ=2π∫0∞f~λhsin⁡λx-λcos⁡λxh2+λ2dλagain subject to the existence of the integral used in the definition.Proof.
Let functionf(x) take the following form:(7)fx=2π∫0∞f~λhsin⁡λx-λcos⁡λxh2+λ2dλ.Then,(8)hfx+f′x=2π∫0∞f~λsin⁡λxdλ.Due to the inverse Fourier cosine transform, we get(9)f~λ=∫0∞sin⁡λxhfx+f′xdλ=sin⁡λxfx0∞+∫0∞hsin⁡λx-λcos⁡λxfxdx=∫0∞hsin⁡λx-λcos⁡λxfxdx.The 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:(10)-ddxAxddxφx,λ=λ2φx,λ,x∈In+,where (11)φx,λ=∑j=1n+1θx-lj-1-θx-ljφjx,λ,bounded nontrivial unknown matrix function of size r×r called matrix eigenfunction of Sturm-Liouville problem, Ax is the matrix-valued function of size r×r, and(12)In+=x:x∈Uj=1n+1lj-1,lj,l0=0,ln+1=∞.In general, Sturm-Liouville matrix problem does not possess an analytical solution. Therefore, we consider the Sturm-Liouville piecewise approximation as follows.
A x is piecewise constant; that is,(13)Ax=∑j=1n+1Ai2θx-lj-1-θx-lj,where x=lj,j=1,…,n+1, are points of discontinuity in In+, and θx is the Heaviside step function.
The elements of matrix eigenfunctionsφ(x,λ) of a Sturm-Liouville matrix problem are piecewise trigonometric functions. The explicit expression of spectral matrix-valued function φ(x,λ) 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 y bounded on the set In+(14)yx,λ=∑k=1n+1θx-lk-1θlk-xykx,λ,to a system of an ordinary differential equations with constant matrix coefficients(15)Am2d2dx2+λ2Eym=0,m=1,n+1¯with the boundary conditions at the points l0 and ln+1,(16)α110+λ2δ110ddx+β110+λ2γ110y1=0,(17)yn+1x=∞<∞,and internal boundary conditions at the points x=lk(18)αj1k+λ2δj1kddx+βj1k+λ2γj1kyk=αj2k+λ2δj2kddx+βj2k+λ2γj2kyk+1,where (19)αjik,βjik,γjik,δjik,Ajare matrices of size r×r:(20)ymx,λ=y1mx,λ⋮yrmx,λ,ym=y1m2+⋯+yrm2.Let y be nontrivial solution to boundary value problems (52)–(69) for some λ. The number λ is called an eigenvalue, and the corresponding solution yx,λ is called matrix-valued eigenfunction.We will required invertible conditions(21)det⁡Mmk≠0,λ∈0,∞for matrices(22)Mmk≡β1mk+λ2γ1mkα1mk+λ2δ1mkβ2mk+λ2γ2mkα2mk+λ2δ2mk.The matrices Am2 are positive-defined [4]. We denote(23)Φn+1x=eqn+1xi;Ψn+1x=e-qn+1xi,qn+12=λ2An+1-2.We define the other n pairs of matrix-value functions Φk,Ψk,k=1,n¯ by the induction relations:(24)αj1k+λ2δj1kddx+βj1k+λ2γj1kΦkΨk=αj2k+λ2δj2kddx+βj2k+λ2γj2kΦk+1Ψk+1.Introduce the following notations:(25)Φ0,1λ=α110+λ2δ110Φ1′l0,λ+β110+λ2γ110Φ1l0,λ,Ψ0,1λ=α110+λ2δ110Ψ1′l0,λ+β110+λ2γ110Ψ1l0,λ,Ωk=ΦkΨkΦk/Ψk/.Lemma 2.
The following identity(26)Ωk-1x,λ′=0E-qk20Ωk,Ωk-1x,λ′′=-Ωk-1λ2Ak-2holds true for k=1,…,n.Proof.
Using formula [20](27)Ωk-1′=-Ωk-1Ωk′Ωk-1and identity(28)Ωk′=Φk′Ψk′Φk′′Ψk′′=0E-qk20Ωk,we can conclude(29)Ωk-1′=-Ωk-10E-qk20,Ωk-1′′=Ωk-10E-qk200E-qk20=Ωk-1-qk200-qk2.Lemma 3.
If the following inequality(30)det⁡β110+λ2γ110α110+λ2δ1100E≠0,λ∈0,∞holds true, then(31)det⁡Φ0,1λ≠0,det⁡Ψ0,1λ≠0,λ∈0,∞.Proof.
MatricesΦ0,1 and Ψ0,1 are nonsingular due to the following identity:(32)Φ0,1Ψ0,1Φ1′Ψ1′=β110+λ2γ110α110+λ2δ1100EΦ1Ψ1Φ1′Ψ1′.Theorem 4.
The spectrum of problems (15), (16), and (18) is continuous and fills semiaxis 0,∞. Sturm-Liouville problem is r time singular. Exactly r linearly independent matrix-valued functions correspond to each eigenvalue λ. It is possible to take r columns of matrix-value functions:(33)vx,λ=∑k=1n+1θx-lk-1θlk-xvkx,λ,vjx,λ=Φjx,λΦ0,1-1λ-Ψjx,λΨ0,1-1λ.That is(34)ymx,λ=v1mx,λ⋮vrmx,λ.Theorem4 follows from Lemmas 2 and 3.Now we consider the dual matrix Sturm-Liouville problem. We find the nontrivial solutiony∗ of a system of ordinary differential equations with constant matrix coefficients(35)Am2d2dx2+λ2Eym∗=0,m=1,n+1¯,with boundary conditions at the points x=l0 and x=ln+1(36)y1∗α110+λ2δ110-1+ddxy1∗β110+λ2γ110-1=0,(37)yn+1∗<∞,and internal boundary conditions at the points x=lk,(38)-yk∗′yk∗β11k+λ2γ11kα11k+λ2δ11kβ21k+λ2γ21kα21k+λ2δ21k-1=-yk+1∗′yk+1∗β12k+λ2γ12kα12k+λ2δ12kβ22k+λ2γ22kα22k+λ2δ22k-1.We write the solution of the boundary value problem in the following form:(39)y∗ξ,λ=∑k=1n+1θξ-lk-1θlk-ξyk∗ξ,λ,ym∗ξ,λ=ym1∗ξ,λ⋯ymr∗ξ,λ,ym∗=y1m∗2+⋯+yrm∗2.Theorem 5.
The spectrum of problems (35), (36), and (38) is continuous and fills semiaxis 0,∞. Sturm-Liouville problem is r time singular. Exactly r linearly independent matrix-valued functions correspond to each eigenvalue λ. It is possible to take r rows of matrix-value functions: (40)v∗x,λ=∑k=1n+1θx-lk-1θlk-xvk∗x,λ,vj∗x,β=Φ0,1Ψ0,1Ωj-1x,β0EAj-2.That is,(41)y∗jξ,λ=vj1∗ξ,λ⋯vjr∗ξ,λ,j=1,r¯.Theorem5 follows from Lemmas 2 and 3.The explicit expression of spectral matrix-valued functionux,λ and the dual spectral function u∗x,λ allow for writing the decomposition theorem on the set In+.Theorem 6.
Let vector-valued functionf(x) be defined over In+, continuous, absolutely integrable, and has a bounded variation. Then, for any x∈In+, the decomposition formula(42)fx=-1πi∫0∞vx,λ∫l0∞v∗ξ,λfξdξ+γ110f1l0+δ110f1′l0+∑k=1nΦ0,1Ψ0,1Ωk-1lk,λMk1-1λ·γ21kδ21kγ22kδ22kfk+1lkfk+1′lk-γ11kδ11kγ12kδ12kfklkfk′lkλdλholds 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 Theorem6.The direct transform is(43)Fn+fλ=∫l0∞v∗ξ,λfξdξ+γ110f1l0+δ110f1′l0+∑k=1nΦ0,1Ψ0,1Ωk-1lk,λMk1-1λ·γ21kδ21kγ22kδ22kfk+1lkfk+1′lk-γ11kδ11kγ12kδ12kfklkfk′lkand the inverse transform is(44)Fn+-1f~x=-1πi∫0∞λvx,λf~λdλ≡fx,when(45)fx=∑k=1n+1θlk-xθx-lk-1fkx.Now we will get the result of the basic identity of matrix integral transforms with piecewise trigonometric kernels for differential operator:(46)B=∑j=1n+1θx-lj-1θlj-xAj2d2dx2.Theorem 7.
If vector-valued function(47)fx=∑k=1n+1θx-lk-1θlk-xfkxis three times continuously differentiable over set In+, has the limit values of its third-order derivatives at the points x=lk, and satisfies boundary condition on infinity(48)limx→∞⁡u∗x,λddxfx-ddxu∗x,λfx=0and internal boundary conditions at the points x=lk, then the basic identity(49)Fn+Bfλ=-λ2f~λ-β110f1l0+α110f1′l0-γ110A12f1′′l0+δ110A12f1′′′l0-∑k=1nΦ0,1Ψ0,1Ωk-1lk,λMk1-1λ·β21kα21kβ22kα22kfk+1lkfk+1′lk-γ21kδ21kγ22kδ22kAk+12fk+1′′lkAk+12fk+1′′′lk-β11kα11kβ12kα12kfklkfk′lk-γ11kδ11kγ12kδ12kAk2fk′lkAk2fk′′′lkholds 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 condition(50)-A2d2dx2φx,λ=λ2φx,λ,0<x<∞,φ0,λ=0,φx,λ<∞provides the direct and inverse matrix integral sine transforms on the real semiline:(51)F+fλ≡f~λ=∫0∞sin⁡A-1λξfξdξ,fx=2A-1π∫0∞sin⁡A-1λxf~λdλ.Proof.
Performing calculations in formulas (33) and (41), we get (52)Φx,λ=exp⁡iA-1λx,Ψx,λ=exp⁡-iA-1λx,Φ0,1=E,Ψ0,1=E,vx,λ=Φx,λ-Ψx,λ=2isin⁡A-1λx,v∗x,β=EEΦΨΦ′Ψ′-10EA-2=-A2iEE-iA-1λxe-iA-1λx-e-iA-1λx-iA-1λxeiA-1λxeiA-1λx·0EA-2=A-1sin⁡A-1λx.Theorem 9.
The matrix-valued Sturm-Liouville problem with Neumann boundary condition(53)-A2d2dx2φx,λ=λ2φx,λ,0<x<∞,φ′0,λ=0,φx,λ<∞provides the direct and inverse matrix integral cosine transforms on the real semiline:(54)F+fλ≡f~λ=∫0∞cos⁡A-1λξfξdξ,fx=2A-1π∫0∞cos⁡A-1λxf~λdλ.Theorem 10.
The matrix-valued Sturm-Liouville problem with Robin boundary condition(55)-A2d2φdx2=λ2φx,λ,x>0,Hφ+dφdx=0,x=0,φx,λ<∞,where H is the square matrix with negative eigenvalues, provides the direct and inverse matrix Fourier type transforms on the real semiline:(56)F+fλ≡f~λ=∫0∞Hsin⁡A-1λξ-A-1λcos⁡A-1λξAfξdξ,fx=2π∫0∞AHsin⁡A-1λξ-A-1λcos⁡A-1λξ·AHA2+λ2E-1f~λdλ.Proof.
Substitute(57)Φx,λ=exp⁡iA-1λx,Ψx,λ=exp⁡-iA-1λxinto (33) and (41); then, (58)Φ0,1=H+iλA-1,Ψ0,1=H-iλA-1,vx,λ=Φx,λH+iλA-1-1-Ψx,λH-iλA-1-1=AΦx,λHA-iλE-Ψx,λHA+iλE·HA2+λ2E-2=2iAsin⁡λA-1xH-A-1λcos⁡A-1λxA;v∗x,β=EEΦΨΦ′Ψ′-10EA-2=-E2iH+iλA-1H-iλA-1·-e-iA-1λxeiA-1λx0EA-1=-Hsin⁡λA-1x+A-1λcos⁡A-1λxA.Theorem 11.
The matrix-valued Sturm-Liouville problem with Dirichlet boundary condition on the composite semilineI1+=(0,l)∪(l,∞)(59)-A12d2φ1dx2=λ2φ1x,λ,x∈0,l;A22d2φ2dx2=λ2φ2x,λ,x∈l,∞φ1=0,x=0,φ1=φ2,λ1∂xφ1=λ2∂xφ2,x=l,φ2x,λ<∞provides the direct and inverse matrix sine type integral transforms on the composite real semiline:(60)F+fλ≡f~λ=∫0lsin⁡q1ξ·A1-1f1ξdξ+∫l∞cos⁡q1l·sin⁡q2ξ-l+sin⁡q1lχcos⁡q2ξ-lA2-1f2ξdξ,f1x=2π∫0∞sin⁡q1xf~λdλ,f2x=2π∫0∞cos⁡q2x-lsin⁡q1l+sin⁡q2x-lχ-1cos⁡q1lΩf~λdλ,where(61)Ω=sin⁡q1lχsin⁡q1l+cos⁡q1lχ-1cos⁡q1l-1,q1=A1-1λ1,q2=A2-1λ1·χ=λ2A2-1A1λ1-1.Proof.
Performing calculations in formulas (33) and (41), we get(62)Φ2=cos⁡q2x-l+isin⁡q2x-l,x>l,Ψ2=cos⁡q2x-l-isin⁡q2x-l,x>l,Φ1=cos⁡q1x-l+isin⁡q1x-l·χ,0<x<l,Ψ1=cos⁡q1x-l-isin⁡q1x-l·χ,0<x<l,Φ0,1=cos⁡q1l-isin⁡q1l·χ,Ψ0,1=cos⁡q1l+isin⁡q1l·χ.Then, matrix eigenfunctions of Sturm-Liouville problem have the following form:(63)v2=Φ2x,λΦ0,1-1λ-Ψ2x,λΨ0,1-1λ=2icos⁡q2x-lsin⁡q2x-l·cos⁡q1lsin⁡q1l·χ-sin⁡q1l·χcos⁡q1l-10E=2i·cos⁡q2x-lsin⁡q1l+sin⁡q2x-lχ-1cos⁡q1lΩ,v1=Φ1x,λΦ0,1-1λ-Ψ1x,λΨ0,1-1λ=2icos⁡q1x-lsin⁡q1x-lχ-1·cos⁡q1lsin⁡q1l·χ-sin⁡q1l·χcos⁡q1l-10E=2i·cos⁡q2x-lsin⁡q1l+sin⁡q2x-lcos⁡q1lΩ=2isin⁡q1x·Ω.And dual matrix eigenfunctions of Sturm-Liouville problem have the following form:(64)v1∗x,β=Φ0,1Ψ0,1Ω1-1x,β0EA1-2=Φ0,1Ψ0,1·cos⁡q1x-lsin⁡q1x-lχ-q1sin⁡q1x-lχq1cos⁡q1x-l-1·0EA1-2=cos⁡q1lsin⁡q1l·cos⁡q1x-lsin⁡q1x-l·χ-q1sin⁡q1x-l·χq1cos⁡q1x-l-1·0EA1-2=-A1-1λsin⁡q1x,(65)v2∗x,β=Φ0,1Ψ0,1Ω2-1x,β0EA2-2=Φ0,1Ψ0,1·cos⁡q2x-lsin⁡q2x-lχ-q2sin⁡q2x-lχq2cos⁡q2x-l-1·0EA2-2=cos⁡q1lsin⁡q1l100-χ·cos⁡q2x-lsin⁡q2x-l-q1sin⁡q2x-lq1cos⁡q2x-l-1·0EA2-2=-cos⁡q1lsin⁡q2x-l+sin⁡q1l·χcos⁡q2x-lA2-1λ.
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].Let(66)u=∑k=1n+1θx-lk-1θlk-xukt,xbe a solution of system of differential equations(67)∂t-ak2∂xx2uk=0,x∈lk-1,lk,k=1,…,n+1with initial conditions(68)uk0,x=fkx,∂tuk0,x=0with boundary conditions(69)h1u1+∂xu1=0,h2∂xxu1+∂xxxu1=0,ifx=0and internal boundary conditions at the points x=lk(70)uk=uk+1,λk∂xuk=λk+1∂xuk+1,μk∂xxuk=μk∂xxuk+1,νk∂xxxuk=νk+1∂xxxuk+1.At the beginning, we will solve an auxiliary vector mixed boundary value problem. Let (71)uv=∑k=1n+1θx-lk-1θlk-xukvkbe a solution to the system of the differential equations(72)∂tukvk=ak2-ak20ak2∂xxukvk,with initial conditions(73)ukvk=fkxfkx,ift=0with boundary conditions at the point x=0(74)h10h2-h1h2u1v1+∂xu1v1=00and internal boundary conditions at the points x=lk(75)10μkak2μkak2ukvk=10μk+1ak+12μk+1ak+12uk+1vk+1,(76)λ10νkak2νkak2∂xukvk=λ20νk+1ak+12νk+1ak+12∂xuk+1vk+1.Lemma 12.
The solution of problems (72)–(76) has the following form:(77)uv=Fn+-1e-λ2tf~λ,where(78)f~λ=∫0∞v∗ξ,λfξfξdξ.Proof.
This lemma can be proved by Fourier integral transform method with piecewise trigonometric kernels (43)-(44). Let the vector-function(79)u~v~≡Fn+uv=∫0∞v∗ξ,λuξvξdξbe the Fourier transform. Then, from Theorem 7, vector-function u~v~ will be a solution of the Cauchy problem: (80)ddtu~v~=-λ2u~v~,u~0v~0=Fn+fxfx.The solution has the following form:(81)u~v~=e-λ2tFn+fxfx.To complete the proof, we apply inverse Fourier transform Fn+-1:(82)uv=Fn+-1e-λ2tf~λ.Theorem 13.
Letuv be a solution of vector problems (72)–(76); then,(83)uk=10ukvk,k=1,…,n+1is a solution to mixed boundary value problems (67)–(70) for iterated heat conduction equation (67).Proof.
In accordance with (72), the function u is a solution of iterated heat conduction equation (67):(84)∂t-a2∂xx2u=a2∂t∂t-a2∂xxv=0.Due to (76), we get the initial condition: (85)∂tu0,x=a2∂xxu0,x-a2∂xxv0,x=a2f′′x-a2f′′x=0.On the basis of (74), the Robin boundary conditions have the following form: the first condition is as follows:(86)h2-h1u+h2v+∂xv=0,ifx=0,and the second condition is as follows: if x=0, then (87)h2∂xxu+∂xxxu=h2∂tu+∂tva2+∂x∂tu+∂tva2=h2∂tu+∂txu+h2∂tv+∂txva2=h2∂tu+∂txu+h1-h2∂tua2=∂th1u+∂xua2=0.It follows from (93)-(94) that(88)∂tμkuk+vkak2=∂tμk+1uk+1+vk+1ak+12;∂x∂tνkuk+vkak2=∂x∂tνk+1uk+1+vk+1ak+12∂tui+viai2=∂xxui.Then, the internal boundary conditions at the points x=lk(89)μk∂xxuk=μk+1∂xxuk+1,νk∂xxxuk=νk+1∂xxxuk+1hold true.Corollary 14.
The solution of problems (67)–(70) has the following form:(90)u=10uv=10Fn+-1e-λ2tf~λ=10∫0∞vk∗ξ,λ11fkξdξ.


## 5. Stress Produced in the Elastic Semi-Infinite Solid by Pressure

Let us consider a problem about distribution of tension in ann+1-layer elastic semi-infinite solid: (91)In+×R=x,y:x∈In+,y∈R.In the case of plane, the strain vector of displacementu¯i has components ui,vi,0. Introduce Airy stress function [10](92)p=∑k=1n+1θx-lk-1θlk-xpky,xas a solution to system of differential equations(93)∂yy+∂xx2pk=0,x∈lk-1,lk,k=1,…,n+1with boundary conditions(94)σ1x=-p′′y,τ1x=0,ifx=0and internal boundary conditions at the points x=lk(95)σkx=σk+1x,τkxy=τk+1xy,uk=uk+1,vk=vk+1,where σkx is the normal stress and tkxy is the shearing stresses. Fourier transform of Airy stress function with respect to y(96)p¯=∑k=1n+1θx-lk-1θlk-xp¯ky,xis the solution to system of differential equations(97)-ξ2+dxx2p¯k=0,x∈lk-1,lk,k=1,…,n+1with boundary conditions(98)σ¯1x=ξ2p¯ξ,τ¯1x=0,ifx=0and internal boundary conditions at the points x=lk(99)σ¯kx=σ¯k+1x,τ¯kxy=τ¯k+1xy,u¯k=u¯k+1,v¯k=v¯k+1.At the beginning, we will solve an auxiliary vector mixed boundary value problem. Let (100)p¯q¯=∑k=1n+1θx-lk-1θlk-xp¯kq¯kbe a solution to the system of differential equations(101)-ξ2p¯kq¯k+1-101dxxp¯kq¯k=0,with boundary conditions(102)p¯1=-p¯ξ,∂xp¯1=0and internal boundary conditions at the points x=lk(103)101+σkEk-1-σk2Ekp¯kq¯k=101+σk+1Ek+1-1-σk+12Ek+1p¯k+1q¯k+1,(104)101+σkEk1-σk2Ek∂xp¯kq¯k=101+σk+1Ek+11-σk+12Ek+1∂xp¯k+1q¯k+1.Lemma 15.
The solution of problems (101)–(104) in the Fourier images takes the form(105)p¯q¯=-Fn+-11ξ2+λ2p¯ξ0and in the Fourier originals has the form(106)pq=-12π∫0∞Fn+-1e-λy-ηpη0dη.Inverse Fourier transformFn+-1 is constructed in accordance with (44).Theorem 16.
Letp¯q¯ be a solution of vector problems (101)–(104); then,(107)p¯k=10p¯kq¯k,k=1,…,n+1is the solution of scalar problems (75)-(76).Proof.
In accordance with (101), the function p is a solution of iterated Laplace equation (93):(108)-ξ2+dxx2pk=dxx-ξ2+dxxqk=0.Calculating the components u¯k and v¯k of strain vector of displacement on the basis of [10], we get(109)v¯k=1+σk2πE1-σkp¯k′′+σkξ2p¯k=1+σk2πE1-σkξ2p¯k-ξ2q¯k+σkξ2p¯k=1+σk2πEξ2p¯k-1-ξ2q¯k,u¯k=1+σk2πE1-σkp¯k′′′-2-σkξ2p¯k′=1+σk2πE1-σkξ2p¯k′-ξ2q¯k′-2-σkξ2p¯k′=1+σk2πE-ξ2p¯k-1-ξ2q¯k.
As a result, the internal boundary conditions (103)-(104) hold.Corollary 17.
The solution of problems (93)–(95) has the form (110)p=10pq=-1012π∫0∞Fn+-1e-λy-ηpη0dη.


## 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.



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