## 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

## 2. Generalized solutions and conditions on wave fronts

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

where

The matrix

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)

has

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

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

*generalized function* instead of *generalized vector function*).

Let

Let us consider Eq. (1) in the space *generalized solutions* of Eq. (1) (or *solutions in generalized sense)*.

The solution

**Theorem 2.1.***If**(**) is the generalized solution of*Eq. (1)*, then there are next conditions on the jumps of its components and derivatives*:

where *and the velocity**of a wave front**coincides with one of*

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

Here,

If

If the right part of expression (7) is equal to zero, then the function

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

These conditions on the appropriate mobile wave front

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

If

From here, we have

By virtue of it, the condition (9) will be transformed to the kind (6), where

**Corollary.** On the wave fronts

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),

## 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

**Definition 2.**

and next conditions:

Here, by definition,

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

Here,

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

where

There are the following relations of symmetry and homogeneous:

By virtue of strong hyperbolicity characteristic equation,

has

**Theorem 3.1.***If**are unitary roots of*Eq. (4)*, then the Green’s matrix of system*(1)*has form*

*where**is Heaviside’s function.*

**Theorem 3.2.***If**are roots of*Eq. (4)*with multiplicity**then the Green’s matrix of system*(1)*has form*

*Here, the top index in brackets designate the order of derivative on*

So, the construction of a Green’s matrix is reduced to the calculation of integrals on unit sphere. For odd

We notice that if the original of

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

In the case of invariance of Eq. (1) relative to group of orthogonal transformations, a symbol of the operator

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

where

Here,

From Theorem 3.1 follows the support

in

for odd

For example,

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

**Theorem 3.3.***If the symbol of the operator L is presented in form*(18)*and**are simple roots of*Eq. (4), *then**is defined by the formula*(17), *where**looks like*(20).

If

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:

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

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

If Eqs. (1) are invariant, concerning the group of orthogonal transformations, then

### 3.2 The Green’s tensor of elastic medium

For isotropic elastic medium constants, the matrix is equal to

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

where

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 *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 V ̂ , T ̂ , W ̂ , U ̂ s , T ̂ s

For solution of BVP using Green’s matrix

Then, the equation for

From the invariance of the equations for

Is easy to prove [17].

**Theorem 3.4.***For fixed**and**, the vector**is the fundamental solution of system*(1)*corresponding to*

The matrix *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:

which is the primitive of the corresponding matrices with respect to

It is easy to see that

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

The Green’s matrix of the static equations for

By analogy with (22), we define the matrix

Obviously, we have the symmetry relations

Theorem 3.4 implies the following result.

**Corollary.***fundamental solution of the static equations*:

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*

where

*Here,**are continuous and bounded functions on the sphere**, and**are regular functions that are continuous at**. For any*

and for odd

## 4. Statement of the initial BVP

Consider the system of strict hyperbolic equations (1). Assume that

The boundary

It is assumed that

Furthermore,

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

**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*

*the Dirichlet conditions*

*and the Neumann-type conditions*

**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

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

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

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

*Here and below,**are the differentials of the area of**and**, respectively*.

**Proof**. Multiplying (1) by

This equality is integrated over

Here,

where

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

**Corollary.***If**and*

then

is proved in the following theorem [17]:

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

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*

*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

The vector

Theorem 5.2 yields

Since the integrand is positive definite and by the conditions of the theorem, we have

## 6. Analogues of the Kirchhoff and Green’s formulas

Let us assume that

The Heaviside function

Accordingly, for

which is defined on the entire space

Let

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

The primitive of Green’s matrix with respect to

Here and below, the star denotes the complete convolution with respect to

**Theorem 6.1.***If**is a classical solution of the Dirichlet (Neumann) boundary value problem, then the generalized solution**can be represented as the the sum of the convolutions*

*Here,**is a singular generalized function that is a single layer on**(see [*2*]), and**is a single layer on*

**Proof**. Applying the operator

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

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

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

Let us show that

Here,

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).***If**is an arbitrary closed Lyapunov surface in**then*

*For**the integral is singular and is understood in the sense of its principal value*.

**Proof**. Convolution Eq. (27) with

Using (29), we obtain the formula in the lemma. Since

Let

Similarly, we obtain

Since the outward normals to

For

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

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

**Theorem 7.1.***If**is a classical solution of the boundary value problem, then*

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

**Proof**. For even

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

**Remark.** If

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

Here, the first integral is singular for

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

Let

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

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 for**and**satisfies the singular boundary integral equations* (

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 (

## 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).