Open access peer-reviewed chapter - ONLINE FIRST

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

By Lyudmila A. Alexeyeva and Gulmira K. Zakiryanova

Submitted: November 13th 2019Reviewed: April 10th 2020Published: May 27th 2020

DOI: 10.5772/intechopen.92449

Downloaded: 26

Abstract

The method of boundary integral equations is developed for solving the nonstationary boundary value problems (BVP) for strictly hyperbolic systems of second-order equations, which are characteristic for description of anisotropic media dynamics. The generalized functions method is used for the construction of their solutions in spaces of generalized vector functions of different dimensions. The Green tensors of these systems and new fundamental tensors, based on it, are obtained to construct the dynamic analogues of Gauss, Kirchhoff, and Green formulas. The generalized solution of BVP has been constructed, including shock waves. Using the properties of integrals kernels, the singular boundary integral equations are constructed which resolve BVP. The uniqueness of BVP solution has been proved.

Keywords

  • hyperbolic equations
  • generalized solution
  • Green tensor
  • boundary value problem
  • generalized function method

1. Introduction

Investigation of continuous medium dynamics in areas with difficult geometry with various boundary conditions and perturbations acting on the medium leads to boundary value problems for systems of hyperbolic and mixed types. An effective method to solve such problems is the boundary integral equation method (BIEM), which reduces the original differential problem in a domain to a system of boundary integral equations (BIEs) on its boundary. This allows to lower dimension of the soluble equations, to increase stability of numerical procedures of the solution construction, etc. Note that for hyperbolic systems, BIEM is not sufficiently developed, while for solving boundary value problems (BVPs) for elliptic and parabolic equations and systems, this method is well developed and underlies the proof of their correctness. It is connection with the singularity of solutions to wave equations, which involve characteristic surfaces, i.e., wavefronts, where the solutions and their derivatives can have jump discontinuities. As a result, the fundamental solutions on wavefronts are essentially singular, and the standard methods for constructing BIEs typical for elliptic and parabolic equations cannot be used. Therefore, for the development of the BIEM for hyperbolic equations, the theory of generalized functions [1, 2] is used. At present, BIEM are applied very extensively to solve engineering problems.

Here, the second-order strictly hyperbolic systems in spaces of any dimension are considered. The fundamental solutions of consider systems of equations are constructed and their properties are studied. It is shown that the class of fundamental solutions for our equations in spaces of odd dimensions is described by singular generalized functions with a surface support (e.g. for R3×t, this is a single layer on a light cone). The constructed fundamental solutions of consider systems of equations are the kernels of BIEs. For systems of hyperbolic equations, the BIE method is developed. Here, the ideas for solving nonstationary BVPs for the wave equations in multidimensional space [3, 4] are used and the methods were elaborated for boundary value problems of dynamics of elastic bodies [5, 6, 7, 8].

2. Generalized solutions and conditions on wave fronts

Consider the second-order system of hyperbolic equations with constant coefficients:

Lijxtujxt+Gixt=0,xtRN+1E1
Lijxt=Cijmlmlδijt2,i,j=1,M¯,m,l=1,N¯E2
Cijml=Cijlm=Cjiml=CmlijE3

where GiL2RN+1and δijare Kronecker symbols; x=1N, i=/xi,and t=/tare Partial derivatives; and also we will use following notations ui,j=juiand ui,t=tui.

The matrix Cijml, whose indices may be permitted in accordance with above indicated symmetry properties (3), satisfies the following condition of strict hyperbolicity:

Wnv=Cijmlnmnlvivj>0n0,v0

Here everywhere like numbered indices indicate summation in specified limits of their change (so as in tensor convolutions).

By the virtue of positive definiteness W, the characteristic equation of the system (1)

detCijmlnmnlc2δij=0,n=1E4

has 2Mvalid roots (with the account of multiplicity):

c=±ckn:0<ckck+1,k=1,M1¯

They are sound velocities of wave prorogations in physical media which are described by such equations. In a general case, they depend on a wave vector n.

It is known that the solutions of the hyperbolic equations can have characteristic surfaces on which the jumps of derivatives are observed [9]. To receive the conditions on jumps, it is convenient to use the theory of generalized functions.

Denote through DMRN+1the space of generalized vector functions f̂xt=f̂1f̂Mdetermined on the space DMRN+1of finite and indefinitely differentiable vector functions φxt=φ1φM.For regular f̂, this linear function is presented in integral form:

f̂xtφxt=RNfixτφixτdVx,φDMRN+1,i=1,M¯

dV=dx1dxN(further, we shall say everywhere generalized function instead of generalized vector function).

Let uxtbe the solution of Eq. (1) in RN+1, continuous, twice differentiable almost everywhere, except for characteristic surface Fwhich is motionless in RN+1and mobile in RN(wave front Ft). On surface, Ftderivatives can have jumps. The equation of Fis Eq. (4). We denote ν=n1,nNnt=ntt, n=n1,nN, where νis a normal vector to the characteristic surface Fin RN+1, and nis unit wave wave vector in RNdirected in the direction of propagation Ft. It is assumed that the surface Fis piecewise smooth with continuous normal on its smooth part.

Let us consider Eq. (1) in the space DMRN+1and its solutions in this space are named as generalized solutions of Eq. (1) (or solutions in generalized sense).

The solution u(x,t) is considered as a regular generalized function and we denote û(x,t) = u(x,t), accordingly Ĝ(x,t) =G(x,t). Let û(x,t) be the solution of Eq. (1) in DMRN+1.

Theorem 2.1.Ifû(x,t) is the generalized solution ofEq. (1), then there are next conditions on the jumps of its components and derivatives:

uixtFt=0E5
σimnmcui,tFt=0E6

where σim=Cijmluj,land the velocitycof a wave frontFtcoincides with one ofck.

Proof. By the account of differentiation of regular generalized function rules [2], we receive:

Lijxtûjxt+Ĝixt=σimνmνtui,tFδFxt+
+CijmlmujFνlδFxtuiFνtδFxt,tE7

Here, αxtδF(x,t) is singular generalized function, which is a simple layer on the surface Fwith specified density α=α1αM:

αxtδFxtφxt=FαixtφixtdSxt,φxtDMRN+1

dSxtis the differential of the surface in a point xtand (ν,νt) = ν1νNνtis a unit vector, normal to characteristic surface F.

If Fxt=0is an equation of wave front, then

ννt=gradFF,t/gradFF,t.

If the right part of expression (7) is equal to zero, then the function û(x,t) will satisfy to the Eq. (1) in a generalized sense. The natural requirement of the continuity of the solutions at transition through wave front F

uixtF=0E8

vanishes only two last composed right parts of Eq. (7). Hence, it is necessary that

σimνmνtui,tF=0E9

These conditions on the appropriate mobile wave front Ftwe can write down with the account Eq. (4). By virtue of continuity of function u(x,t) for (x,t)Ft, we have

fxtF=limε+0fx+ενt+ενtf(xενtενt)=limε+0fx+εntf(xεnt)=fxtFt;

therefore the condition (5) is equivalent to (8).

If xtFt, then x+cnΔtt+ΔtFt+Δt. Therefore,

Fx+cnΔtt+ΔtFxt=cF,jnj+F,tΔt=0

From here, we have

c=F,t/F,jnj=νt/νiνi

By virtue of it, the condition (9) will be transformed to the kind (6), where c, for each front, coincides with one of ck. The theorem has been proved.

Corollary. On the wave fronts

nlui,t+cui,lFt=0,i=1,M¯,l=1,N¯E10

The proof follows from the condition of continuity (5). The expression (10) is the condition of the continuity of tangent derivative on the wave front.

In the physical problems of solid and media, the corresponding condition (6) is a condition for conservation of an impulse at fronts. This condition connects a jump of velocity at a wave fronts with stresses jump. By this cause, such surfaces are named as shock wave fronts.

Definition 1. The solution of Eq. (1), u(x,t), is named as classical one if it is continuous on RN+1, twice differentiable almost everywhere on RN+1, and has limited number of piecewise smooth wave fronts on which conditions jumps (5) and (6) are carried out.

3. Fundamental matrices

3.1 The Green’s matrix of second-order system of hyperbolic equations

Let us construct fundamental solutions of Eq. (1) on DMRN+1.

Definition 2.Ujkxtis the Green’s matrix of Eq. (1) if it satisfies to equations

LijxtUjkxt+δikδxδt=0,i,j,k=1,M¯E11

and next conditions:

Ujkxt=0fort<0,x,E12
Ujkx0=0forx0E13

Here, by definition,

δikδxtφixt=φk00φDMRN+1

For construction of Green’s matrix, it is comfortable to use Fourier transformation, which brings Eq. (11) to the system of linear algebraic equations of the kind

LjkU¯klξω+δjl=0,j,k,l=1,M¯

Here, ξω=ξ1,,ξN,ωis the Fourier variables appropriate to xt.

By permitting the system, we receive transformation of Green’s matrix which by virtue of differential polynomials uniformity looks like:

U¯jkξω=QjkξωQ1ξωE14

where Qjkare the cofactors of the element with index (k,j) of the matrix L; and Qis the symbol of operator L:

Qξω=detLkj

There are the following relations of symmetry and homogeneous:

Qjkξω=Qjkξω=Qjkξω,Qξω=Qξω=QξωE15
Qjkλξλω=λ2M2Qjkξω,Qλξλω=λ2MQξωE16

By virtue of strong hyperbolicity characteristic equation,

Qξω=0

has 2Mroots. It is a singular matrix. There is not a classic inverse Fourier transformation of it. It defines the Fourier transformation of the full class of fundamental matrices which are defined with accuracy of solutions of homogeneous system (1). Components of this matrix are not a generalized function. To calculate the inverse transformation, it is necessary to construct regularisation of this matrix in virtue of properties (12) and (13) of Green tensor. The following theorems has been proved [10]:

Theorem 3.1.Ifcqq=1,M¯are unitary roots ofEq. (4), then the Green’s matrix of system(1)has form

Ujkxt=σNHtq=1Me=1Ajkecq×ex+cqeti01Nexcqeti01NdSe

whereσN=2πiNN2!,Ajkecq=Qjkecq/2cqQmmecq,and Htis Heaviside’s function.

Theorem 3.2.Ifcqq=1,M¯are roots ofEq. (4)with multiplicitymq, then the Green’s matrix of system(1)has form

Ujkxt=σNHtqmqRNQjk,ωmq1ecqQ,ωmqecq1×ex+cqeti01Nexcqeti01NdSe

Here, the top index in brackets designate the order of derivative onω.

So, the construction of a Green’s matrix is reduced to the calculation of integrals on unit sphere. For odd N, these theorems allow to build the Green’s matrix ε-approach only. For even Nand for εapproach, it is required to integrate multidimensional surface integral over unit sphere. However, in a number of cases, this procedure can be simplified.

We notice that if the original of Q1is known, i.e.

Jxt=F1Q1ξω,

which is built in view of conditions (12), then it is easy to restore the Green’s matrix

Ujkxt=QjkixitJxtE17

In the case of invariance of Eq. (1) relative to group of orthogonal transformations, a symbol of the operator Lijis a function of only two variables ξ,ωand can be presented in the form:

Qξω=2Mqξω1.E18

It essentially simplifies the construction of the original using the Green’s functions of classical wave equations. For this purpose, it is necessary to spread out Q1ξωon simple fractions. In the case of simple roots,

Qξω=k=1Mξ2ω2/ck2
Q1ξω=2M+2k=1MAkξ2ω2/ck21E19

where Akis the decomposition constant. It is easy to see that summand in round brackets under summation sign is the symbol of the classical wave operator

Dk=ck2t2ΔN.

Here, ΔNis the Laplacianfor which the Green’s function UNxthas been investigated well [11].

From Theorem 3.1 follows the support UNxtcis:

Kc+=xt:xctt>0

in RN+1for even Nand it is sound cone

Kc=xt:x=ctt>0

for odd N.

For example, U3is the simple layer on a cone [10] and it is the singular generalized function. In this case, Jxtis convolution over tGreen’s function with Ht:

Jxt=k=1MAkHttHttUN(xtck).E20

Here, the convolution over t undertakes (2 M − 2) time, which exists, by virtue of, on semi-infinite at the left of supports of functions [11]. It is easy to check up that the boundary conditions (12) and (13) are carried out as UNxtcwhich satisfies them. We formulate this result as:

Theorem 3.3.If the symbol of the operator L is presented in form(18)andckare simple roots ofEq. (4), thenUjkxtis defined by the formula(17), whereJxtlooks like(20).

If ckhave multiplicity mkin decomposition as (20), degrees ξ2ω2/ck2mm=1,mk¯can appear. Using the property of convolution transformation, we receive their original in kind of complete convolution over (x,t):

F1ξ2ω2/ck2m=UNxtcmUN(xtc)

Then, the procedure of construction of a Green’s matrix is similar to the described one.

We notice that as follows from (20) in a case of N = 1, 2, the convolution operation is reduced to calculate regular integrals of simple kind:

UNxttHt=0tUNxtτ
UNxttUNxt=RNdVy0tUNxytτUNyτ

But already for N = 3 and more, the construction of convolutions is non-trivial, and for their determination, its definition in a class of generalized functions should be used.

For any regular function ĜDMRN+1:suptĜ0, the appropriate solution of Eq. (1) looks like the convolution

ûi=UikĜk.

For regular functions, it has integral representation in form of retarded potential:

ûixt=Ht0RNUikxyτGkytτdVy

If Eqs. (1) are invariant, concerning the group of orthogonal transformations, then ckdo not depend on n. In physical problems, the isotropy of medium is reduced to the specified property.

3.2 The Green’s tensor of elastic medium

For isotropic elastic medium constants, the matrix is equal to

Cijml=ρλδlmδij+μδimδjl+δjmδil.

The coefficients of Eq. (1) depend only on two sound velocities

c1=λ+2μ/ρ,c2=μ/ρ,

where ρis the density of medium, and λand μare elastic Lame parameters. These two speeds are velocities of propagation of dilatational and shearing waves. Wave fronts for Green’s tensor are two spheres expanding with these velocities.

In the case of plane deformation N = M = 2, an appropriate Green’s tensor was constructed in [5, 6]. For the space deformation N = M = 3, the expression of a Green’s tensor was represented in [6].

For anisotropic medium in a plane case (N = M = 2), the Green’s tensor was constructed in [12, 13]. For such medium, the wave propagation velocities depend on direction nand the form of wave fronts essentially depends on coefficients of Eq. (1). Anisotropic mediums with weak and strong anisotropy of elastic properties in the case of plane deformation were considered in [12, 13, 14, 15]. In the first case, the topological type of wave fronts is similar to extending spheres. In the second case, the complex wave fronts and lacunas appear [16]. Lacunas are the mobile unperturbed areas limited by wave fronts and extended with current of time. Such medium has sharply waveguide properties in the direction of vector of maximal speeds. The wave fronts and the components of Green’s tensor for weak and strong anisotropy are presented in [15]. The calculations are carried out for crystals of aragonite, topaz and calli pentaborat.

3.3 The fundamental matrices V̂,T̂,Ŵ,Ûs,T̂s

For solution of BVP using Green’s matrix Û, we introduce the fundamental matrices Ŝand T̂with elements given by

Ŝikmxt=CijmllÛjk,Γikxtn=Ŝikmnm,E21
T̂kixtn=Γikxtn=CijmlnmlÛjk,E22
i,j,k=1,M¯,m,l=1,N¯.

Then, the equation for Ûcan be written as

Ŝikl,lÛik,tt+δikδxδx=0.

From the invariance of the equations for Ûunder the symmetry transformations y=x, some symmetry properties of introduced matrices follows:

Ûikxt=Ûikxt,Ûikxt=Ûkixt,Ŝikmxt=Ŝikmxt,E23
T̂ikxtn=T̂ikxtn=T̂ikxtn.E24

Is easy to prove [17].

Theorem 3.4.For fixedkandn, the vectorT̂ikxtnis the fundamental solution of system(1)corresponding to

Gi=Cikmlnmδ,lxδt.

The matrix T̂is called a multipole matrix, since it describes the fundamental solutions of system (1) generated by concentrated multipole sources (see [18]).

Primitives of the matrix. The primitive of the multipole matrix is introduced as convolution over time:

Ŵjkxtn=T̂jkxtntHt,

which is the primitive of the corresponding matrices with respect to t:

tV̂ik=Ûikxt,tŴik=T̂ikxtn.

It is easy to see that V̂ikand Ŵikare fundamental solutions to system (1) of the form

LijxtV̂jk+δikδxHt=0,E25
LijxtŴjk+nmCkimlδ,lxHt=0.

Relation (23) implies the following symmetry properties of the above matrices:

V̂ikxt=V̂ikxt,V̂ikxt=V̂kixt,
Ŵikxtn=Ŵikxtn=Ŵikxtn.E26

The Green’s matrix of the static equations for Ûiksx(when the t-derivatives in (1) are zero) is defined by

Lijx0Ûjksx+δikδx=0,E27
Ûiksx0,x.E28

By analogy with (22), we define the matrix

T̂iksxn=CkjmlnmlÛjis.

Obviously, we have the symmetry relations

T̂iksxn=T̂iksxn=T̂iksxn.E29

Theorem 3.4 implies the following result.

Corollary.T̂iksis a fundamental solution of the static equations:

Lijx0TjksnmCkimlδ,lx=0.

It is easy to see that this is an elliptic system.

The following theorem have been proved [17].

Theorem 3.5.The following representations take place

V̂ikxt=UiksxHt+Vikdxt,E30
Ŵikxt=TiksxHt+Wikdxt,E31

where UiksxHtand TiksxHtare regular functions for x0. As x0,

UiksxlnxAikNex,Tiksxx1BikNex,N=2,
UiksxxN+2AikNex,TiksxxN+1BikNex,N>2.E32

Here,ex=x/x,AikNe,andBikNeare continuous and bounded functions on the spheree=1, andVikdandWikdare regular functions that are continuous atx=0andt>0. For anyN,

Vikdxt=0Wikdxt=0forx>maxk=1,M¯maxe=1cket,

and for odd N, these relations hold for x<mink=1,M¯mine=1cket..

4. Statement of the initial BVP

Consider the system of strict hyperbolic equations (1). Assume that xSRN, where Sis an open bounded set; xtD,D=S×0,Dt=S×0t,t>0; D=S×0, and Dt=S×0t.

The boundary Sof Sis a Lyapunov surface with a continuous outward normal nxn=1:

nx2nx1=Ox2x1β,β>0,x1S,x2S.

It is assumed that Gis a locally integrable (regular) vector function.

G0ast+,xS.

Furthermore, uCD+D, where uis a twice differentiable vector function almost everywhere on D, except for possibly the characteristic surfaces (F) in RN+1, which correspond to the moving wavefronts (Ft) RN. On them, conditions (5) and (6) are satisfied.

It is assumed that the number of wavefronts is finite and each front is almost everywhere a Lyapunov surface of dimension N1.

Problem 1. Find a solution of system (1) satisfying conditions (5)(7) if the boundary values of the following functions are given:

the initial values

uix0=ui0x,xS+SE33
ui,tx0=ui1x,xS;E34

the Dirichlet conditions

uixt=uiSxt,xS,t0;E35

and the Neumann-type conditions

σilxtnlx=gixt,xS,t0,i=1,N¯.E36

Problem 2. Construct resolving boundary integral equations for the solution of the following boundary value problems.

Initial-boundary value problem I. Find a solution of system (1) that satisfies boundary conditions (33)(35) and front conditions (5)(7).

Initial-boundary value problem II. Find a solution of system (1) that satisfies boundary conditions (33), (34), and (36) and front conditions (5)(7).

These solutions are called classical.

Remark. Wavefronts arise if the initial and boundary data do not obey the compatibility conditions

wix0=ui0x,uiS,tx0=ui1x,xS.

In physical problems, they describe shock waves, which are typical when the external actions (forces) have a shock nature and are described by discontinuous or singular functions.

5. Uniqueness of solutions of BVP

Define the functions

Wu=0,5Cijmlui,muj,l,Ku=0,5u,t2,
Eu=Ku+Wu,Lu=KuWu,

which are called the densities of internal, kinetic, and total energy of the system, respectively, and Lis the Lagrangian.

Theorem 5.1.Ifuis a classical solution of the Dirichlet (Neumann) boundary value problem, then

DtLuxtdVxt=DtGixtuixtdVxt+
+DtgixtuiSxtdSxtSuixtui,txtui0xui1xdVx

Here and below,dVx=dx1dxN,dVxt=dVxdt;dSx,and,dSxtare the differentials of the area ofSandD, respectively.

Proof. Multiplying (1) by uiand summing the result over i, after simple algebra, we obtain the expression

L=Cijmluj,mui,luiui,t,t+Giui.

This equality is integrated over Dttaking into account the front discontinuities and using the Gauss-Ostrogradsky theorem and initial conditions (33) and (34) to obtain

DtLuxtdVxt=DtCijmluj,mui,luiui,t,tdVxt++DtGixtuixtdVxt=DtσilnlxuixtdSxtSuiui,txtuiui,tx0)dSxt+DtGixtuixtdVxt++kFkDtuiνlkσilxtνtkui,txtFkdFkxt

Here, νlk,andνtkare the components of the unit normal vector to the front Fkxtin RN+1, for which we have [17]

νtk=ck/νjkνjk1/2,E37

where ckis the velocity of the front. With the notation introduced, the relation (37) and the front condition (7) yield the assertion of the theorem.

It is easy to see that the following result holds true.

Corollary.Ifuix0=0,ui,tx0=0, and

limt+ui,l0,limt+ui,t0,xS,

then

DLuxtdVxt=DGixtuixtdVxt+DgixtuiSxtdSxt

is proved in the following theorem [17]:

Theorem 5.2.Ifuis a classical solution of the Dirichlet (Neumann) boundary value problem, then

SEutEu0dVx=
DtGixtui,txtdVxt+DtgixtuiS,txtdSxt.

It is easy to see that this theorem implies the uniqueness of the solutions to the initial-boundary value problems in question.

Theorem 5.3.If a classical solution of the Dirichlet (Neumann) boundary value problem exists and satisfies the conditions

limt+ui,l0,limt+ui,t0,xS,

then this solution is unique.

Proof. Since the problem is linear, it suffices to prove the uniqueness of the solution to the homogeneous boundary value problem. If there are two solutions u1and u2, then their difference u=u1u2satisfies the system of equations with G=0and the zero initial conditions, i.e.

uimx=0m=01.

The vector uon the boundary Ssatisfies the homogeneous boundary conditions

uixt=0orgixt=0.

Theorem 5.2 yields

SEutdSx=SKut+WutdSx=0.

Since the integrand is positive definite and by the conditions of the theorem, we have u0. The theorem is proved.

6. Analogues of the Kirchhoff and Green’s formulas

Let us assume that Sis a smooth boundary with a continuous normal of a set S. The characteristic function HSxof a set Sis defined for xSas

HSx=1/2E38

The Heaviside function Htis extended to zero by setting H0=1/2. Define the characteristic function of Das

HDxt=HSxHtE39

Accordingly, for udefined on D, we introduce the generalized function

ûxt=uHDxt,E40

which is defined on the entire space RN+1. Similarly,

Ĝkxt=GkHDxt.E41

Let Ûikxtdenotes the Green’s matrix, i.e. the fundamental solution of Eq. (1) that corresponds to the function Fi=δikδxδtand satisfies the conditions

Ûikx0=0,Ûik,tx0=0,x0E42

For system (1), such a matrix was constructed in [10].

The primitive of Green’s matrix with respect to tis defined as

V̂ikxt=ÛikxttHttV̂ik=Ûik.E43

Here and below, the star denotes the complete convolution with respect to xt, while the variable under the star denotes the incomplete convolution with respect to xor t, respectively. The convolution exists since the supports are semibounded with respect to t. Clearly, the convolution is the solution of Eq. (1) at Fi=δikδxHt.

Theorem 6.1.Ifuxtis a classical solution of the Dirichlet (Neumann) boundary value problem, then the generalized solutionûcan be represented as the the sum of the convolutions

ûi=UikĜk+Uikxuk1xHSx+
+tUik*xuk0xHSx+UikgkxtδSxHtE44
CkjmllVikuj,txtnmxδSxHtCkjmllVik*xuj0xnmxδSx.

Here,δSis a singular generalized function that is a single layer onS(see [2]), andgkxtδSxHtis a single layer onD.

Proof. Applying the operator Lijto ûxt, using the differentiation rules for generalized functions, and taking into account the equalities

jHD=njδSxHt,tHD=δtHSx,

and the front conditions (5) and (6), we obtain

Lkjxtûjxt=Ĝkxt+uk1xHSxδt+
+uk0xHSxδ·t+gkxtδSxHtCkjmlujxtnmxδSxHt,l

Next, we use the properties of Green’s matrix to construct a weak solution of Eq. (1) in the form of the convolution

ŵixt=UikĜk+Ûik*xuk1xHSx+tÛik*xuk0xHSx+
+ÛikgkxtδSxHtCkjmlÛikujxtnmxδSxHt,l.E45

The last convolution can be transformed using the relation (43) and applying the differentiation rules for convolutions and generalized functions:

CkjmltV̂ikujnmxδSxHt,l=CkjmllV̂ikujnmxδSxHt,t=
=CkjmllV̂ikuj,tnmxδSxHt+uj0xnmxδSxδt=
=CkjmltV̂ikuj,tnmxδSxHt+CkjmllV̂ik*xuj0xnmxδSx

Let us show that ŵixt=ûixt. Indeed, φDNRN+1

ŵiφi=ÛikF̂kφi=ÛikLkjxtûjφi=
=LkjxtÛikûjφi=δijδxtûjφi=ûiφi.

Here, F̂kdenotes the right-hand side of (44). Since ûiφi=0, if suppφD+, it follows that ŵixt=0,xD. This implies the assertion of the theorem, since the solution of the problem is unique.

Given initial and boundary values (33)(36), the above formula recovers the solution in the domain. For this reason, it can be called an analogue of the Kirchhoff and Green formulas for solutions of hyperbolic systems (1). It gives a weak solution of the problems.

To represent this formula in integral form and use it for the construction of boundary integral equations for solutions of the initial-boundary value problems, we examine the properties of the functional matrices involved.

7. Singular boundary integral equations

Lemma 7.1 (analogue of the Gauss formula).IfSis an arbitrary closed Lyapunov surface inRN, then

STkisyxnydSy=δkiHSx

ForxS, the integral is singular and is understood in the sense of its principal value.

Proof. Convolution Eq. (27) with HSxand using the differentiation rules for convolutions yields

Lijx0UjksHSx+δkiHSx=
=CijmlUjks,lnmδikHSx=STkisxynydSy+δkiHSx=0

Using (29), we obtain the formula in the lemma. Since Tkisis regular for xS, the formula holds for such x. Let us prove the validity of this formula for boundary points.

Let xS. Define Oεx=yS:yx<ε,Sεx=SOεx,Γεx=y:yx=ε,Γεx=ΓεxS,andΓε+x=ΓεxS+.

Similarly, we obtain

SεTkisyxnydSy+ΓεTkisyxnydSy=0
SεTkisyxnydSy+Γε+TkisyxnydSy=δki

Since the outward normals to Γεxand Γε+xat opposite points yand y+of the sphere Γεxcoincide, i.e. ny=xy/ε=y+x/ε=ny+, while y+x=yx, we take into account the asymptotics of Tkisand, according to Theorem 3.5, sum these two equalities and pass to the limit as ε0, to obtain equality (30) for boundary points. The lemma is proved.

For M=1and L1jx0=jj=Δ, this formula coincides with the Gauss formula for the double-layer potential of Laplace equation (see [2]).

Consider formula (44). Formally, it can be represented in the integral form

ûkxt=DTkixynytτuiyt+UkixytτgiyτdDyτ+
+SUki,txytui0y+Uki(xyt)ui1ydVy+UkiĜi

Under zero initial conditions, this formula coincides in form with the generalized Green formula for elliptic systems. However, the singularities of Green’s matrix of the wave equations prevent us from using it for the construction of solutions to boundary value problems, since the integrals on the right-hand side do not exist because Tkihas strong singularities on the fronts. However, the primitives of the matrix introduced in Section 3 can be used to construct integral representations of formula (44).

Theorem 7.1.Ifuis a classical solution of the boundary value problem, then

ûk=UkixtGixt+UkixtgixtδsxHtSTkisxyuiytdSySdSy0tWkidxynytτui,tyτSWkidxynytui0ydSy+Ukixtxui0yHSx,t

ForxS, the integral is singular and is understood in the sense of its principal value.

Proof. For even N, the integral representation (42) has the form

ûk=SdSy0tUkixytτgiyτWkixynytτui,τyτ
SWkixynytui0ydSy+tSUkixytui0ydSy+
+SUkixytui1ydVy+DUkixytτGiyτdVy

Here, all the integrals are regular for interior points and singular for boundary points.

Remark. If Nis odd, then, since Uis singular, the integrals involving Uare still written in the form of a convolution, which is taken according to the convolution rules depending on the form of U. For the wave equation of odd dimension, such representations were constructed in [4].

It is easy to see that, for zero initial data, the last three integrals (in the convolution) vanish.

Applying Theorem 3.5, by virtue of (31), the second term can be represented as

SdSy0tWkixynytτdτuiyτ==STkisxyuiytui0ydSy+SdSy0tWkidxynytτui,τyτ

Here, the first integral is singular for xSand exists in the sense of its principal value by Lemma 7.1, while the second integral is regular. Then for interior points, we obtain the formula of the theorem.

Let us show that the equality holds in the sense of definition (37) for boundary points as well.

Let xS, xSand xx. Then, since the convolutions containing Ukiand Wkidare continuous, we obtain

limxxukxt=ukxt==limxxSTkisyxuiytdSy+SWkidxynytui0ydSySdSy0tUkixytτgi(yτ)+Wkid(xynytτ)ui,τ(yτ)++SUkixytui1ydVy+SUkixytui0y,tdVy++DUkixytτGiyτdVy

By Lemma 7.1, the limit on the right-hand side can be transformed into

STkisyxuiytuixtdSy+uixtδki=
=V.P.STkisyxuiytdSyuixtV.P.STkisyxdSy+
+uixtδki=V.P.STkisyxuiytdSy+0,5uixtδki

Adding up and combining like terms, we derive the formula of the theorem for boundary points. The theorem is proved.

The formula on the boundary yields boundary integral equations for solving initial-boundary value problems.

Theorem 7.2.The classical solution of the Dirichlet (Neumann) initial-boundary value problem forxSandt>0satisfies the singular boundary integral equations (k=1,M¯)

0,5ukxt=UkixtGixt+UkixtgixtδsxHt
V.P.STkisxyuiytdSySdSy0tWkidxynytτui,tyτ
SWkidxynytui0ydSy+Ukixt*xui0yHSx,t+Uik*xuk1xHSx.

From these equations, we can determine the unknown boundary functions of the corresponding initial-boundary value problem. Next, the formulas of Theorem 7.1 are used to determine the solution inside the domain.

8. Conclusions

The solvability of the obtained systems of BIEs in a particular class of functions is an independent problem in functional analysis. These equations can be numerically solved using the boundary element method. In special cases of nonstationary boundary value problems in elasticity theory (M=N=2,3), these equations were solved in [4, 6, 7, 8].

Acknowledgments

This research is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (No. AP05132272, AP05135494 and 3487/GF4).

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Lyudmila A. Alexeyeva and Gulmira K. Zakiryanova (May 27th 2020). Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics [Online First], IntechOpen, DOI: 10.5772/intechopen.92449. Available from:

chapter statistics

26total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us