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

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.


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 R 3 Â 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].

Generalized solutions and conditions on wave fronts
Consider the second-order system of hyperbolic equations with constant coefficients: and ∂ t ¼ ∂=∂t are Partial derivatives; and also we will use following notations The matrix C ml ij , whose indices may be permitted in accordance with above indicated symmetry properties (3), satisfies the following condition of strict hyperbolicity: 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 2M valid roots (with the account of multiplicity): 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 we shall say everywhere generalized function instead of generalized vector function).
Let u x, t ð Þ be the solution of Eq. (1) in R Nþ1 , continuous, twice differentiable almost everywhere, except for characteristic surface F which is motionless in R Nþ1 and mobile in R N (wave front F t ). On surface, F t derivatives can have jumps. The equation of F is Eq. (4). We denote ν ¼ n 1 , … nN , n t ð Þ¼ n, t t ð Þ, n ¼ n 1 , … nN ð Þ , where ν is a normal vector to the characteristic surface F in R Nþ1 , and n is unit wave wave vector in R N directed in the direction of propagation F t . It is assumed that the surface F is piecewise smooth with continuous normal on its smooth part.
Let us consider Eq. (1) in the space D 0 M R Nþ1 À Á and 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 denotê is the generalized solution of Eq. (1), then there are next conditions on the jumps of its components and derivatives: where σ m i ¼ C ml ij u j , l and the velocity c of a wave front F t coincides with one of c k . Proof. By the account of differentiation of regular generalized function rules [2], we receive: Here, α x, t ð Þδ F (x,t) is singular generalized function, which is a simple layer on the surface F with specified density α ¼ α 1 Þis a unit vector, normal to characteristic surface F.
vanishes only two last composed right parts of Eq. (7). Hence, it is necessary that These conditions on the appropriate mobile wave front F t we can write down with the account Eq. (4). By virtue of continuity of function u(x,t) for (x,t) ∈ F t , we have therefore the condition (5) is equivalent to (8).
By virtue of it, the condition (9) will be transformed to the kind (6), where c, for each front, coincides with one of c k . The theorem has been proved.
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), u(x,t), is named as classical one if it is continuous on R Nþ1 , twice differentiable almost everywhere on R Nþ1 , and has limited number of piecewise smooth wave fronts on which conditions jumps (5) and (6) are carried out.

The Green's matrix of second-order system of hyperbolic equations
Let us construct fundamental solutions of Eq.
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 is the Fourier variables appropriate to x, t ð Þ. By permitting the system, we receive transformation of Green's matrix which by virtue of differential polynomials uniformity looks like: where Q jk are the cofactors of the element with index (k, j) of the matrix L Àiξ, Àiω ð Þ f g ; and Q is the symbol of operator L : There are the following relations of symmetry and homogeneous: By virtue of strong hyperbolicity characteristic equation, 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]: Here, the top index in brackets designate the order of derivative on ω. 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 εapproach only. For even N and for εÀapproach, 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 Q À1 is known, i.e.
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 L ij is a function of only two variables ξ k k, ω and can be presented in the form: 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 Q À1 ξ, ω ð Þon simple fractions. In the case of simple roots, where A k is the decomposition constant. It is easy to see that summand in round brackets under summation sign is the symbol of the classical wave operator Here, Δ N is the Laplacian for which the Green's function U N x, t ð Þ has been investigated well [11].
From Theorem 3.1 follows the support U N x, t, c ð Þis: for even N and it is sound cone For example, U 3 is the simple layer on a cone [10] and it is the singular generalized function. In this case, J x, t ð Þ is convolution over t Green's function with H t ð Þ: Here, the convolution over t undertakes (2 M À 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) 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 functionĜ ∈ D 0 M R Nþ1 À Á : sup tĜ ∈ 0, ∞ ð Þ, the appropriate solution of Eq. (1) looks like the convolution For regular functions, it has integral representation in form of retarded potential: If Eqs. (1) are invariant, concerning the group of orthogonal transformations, then c k do not depend on n. In physical problems, the isotropy of medium is reduced to the specified property.

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 ρ is the density of medium, and λ and μ are 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 n and 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. For solution of BVP using Green's matrixÛ, we introduce the fundamental matricesŜ andT with elements given bŷ Then, the equation forÛ can be written aŝ From the invariance of the equations forÛ under the symmetry transformations y ¼ Àx, some symmetry properties of introduced matrices follows: Is easy to prove [17].
Theorem 3.4. For fixed k and n, the vectorT k i x, t, n ð Þis the fundamental solution of system (1) corresponding to The matrixT is 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: which is the primitive of the corresponding matrices with respect to t: It is easy to see thatV Relation (23) implies the following symmetry properties of the above matrices: By analogy with (22), we define the matrix Obviously, we have the symmetry relationŝ Theorem 3.4 implies the following result.

Corollary.T k s ð Þ i
is a 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 placê where Here

Statement of the initial BVP
Consider the system of strict hyperbolic equations (1).
The boundary S of S À is a Lyapunov surface with a continuous outward normal n x ð Þ ∥n∥ ¼ 1 ð Þ : It is assumed that G is a locally integrable (regular) vector function.
G ! 0 as t ! þ∞, ∀x ∈ S À : where u is a twice differentiable vector function almost everywhere on D À , except for possibly the characteristic surfaces (F) in R Nþ1 , which correspond to the moving wavefronts (F t ) R N . 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 N À 1.
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.
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.

Define the functions
which are called the densities of internal, kinetic, and total energy of the system, respectively, and L is the Lagrangian.
Theorem 5.1. If u is a classical solution of the Dirichlet (Neumann) boundary value problem, then ð Here and below, dV ; dS x ð Þ, and, dS x, t ð Þ are the differentials of the area of S and D, respectively.
Proof. Multiplying (1) by u i and summing the result over i, after simple algebra, we obtain the expression This equality is integrated over D t taking into account the front discontinuities and using the Gauss-Ostrogradsky theorem and initial conditions (33) and (34) to obtain Here, ν k l , and ν k t are the components of the unit normal vector to the front F k x, t ð Þ in R Nþ1 , for which we have [17] where c k is 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.
is proved in the following theorem [17]: Theorem 5.2. If u 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. and u 2 , then their difference u ¼ u 1 À u 2 satisfies the system of equations with G ¼ 0 and the zero initial conditions, i.e.
The vector u on the boundary S satisfies the homogeneous boundary conditions u i x, t ð Þ ¼ 0 or g i x, t ð Þ ¼ 0: Since the integrand is positive definite and by the conditions of the theorem, we have u 0. The theorem is proved.

Analogues of the Kirchhoff and Green's formulas
Let us assume that S is a smooth boundary with a continuous normal of a set S À . The characteristic function H À S x ð Þ of a set S À is defined for x ∈ S as The Heaviside function H t ð Þ is extended to zero by setting H 0 ð Þ ¼ 1=2. Define the characteristic function of D À as Accordingly, for u defined on D À , we introduce the generalized function which is defined on the entire space R Nþ1 . Similarly, LetÛ k i x, t ð Þ denotes the Green's matrix, i.e. the fundamental solution of Eq. (1) that corresponds to the function F i ¼ δ k i δ x ð Þδ t ð Þ and satisfies the conditionŝ For system (1), such a matrix was constructed in [10]. The primitive of Green's matrix with respect to t is defined aŝ Here and below, the star denotes the complete convolution with respect to x, t ð Þ, while the variable under the star denotes the incomplete convolution with respect to x or t, respectively. The convolution exists since the supports are semibounded with respect to t. Clearly, the convolution is the solution of Eq. (1) at To represent this formula in integral form and use it for the construction of