**2. Generalized solutions and conditions on wave fronts**

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

$$L\_{ij}(\partial\_{\mathbf{x}}, \partial\_t)u\_j(\mathbf{x}, t) + G\_i(\mathbf{x}, t) = \mathbf{0}, \quad (\mathbf{x}, t) \in \mathbb{R}^{N+1} \tag{1}$$

$$L\_{\vec{\eta}}(\partial\_{\mathbf{x}}, \partial\_{\mathbf{t}}) = \mathbf{C}\_{\vec{\eta}}^{ml} \partial\_{m} \partial\_{l} - \delta\_{\vec{\eta}} \partial\_{\mathbf{t}}^{2}, \quad i, j = \overline{1, M}, \quad m, l = \overline{1, N} \tag{2}$$

$$\mathbf{C}\_{ij}^{ml} = \mathbf{C}\_{ij}^{lm} = \mathbf{C}\_{ji}^{ml} = \mathbf{C}\_{ml}^{j} \tag{3}$$

differentiable vector functions *<sup>φ</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>φ</sup>*<sup>1</sup> ð Þ , … , *<sup>φ</sup><sup>M</sup> :* For regular ^*f*, this linear

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations…*

*dV* ¼ *dx*<sup>1</sup> … *dxN* (further, we shall say everywhere *generalized function* instead of

almost everywhere, except for characteristic surface *F* which is motionless in *RN*þ<sup>1</sup> and mobile in *R<sup>N</sup>* (wave front *Ft*). On surface, *Ft* derivatives can have jumps. The equation of *F* is Eq. (4). We denote *ν* ¼ ð Þ¼ *n*1, … *nN*, *nt* ð Þ *n*, *tt* , *n* ¼ ð Þ *n*1, … *nN* ,

wave vector in *R<sup>N</sup>* directed in the direction of propagation *Ft*. It is assumed that the

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

**Theorem 2.1.** *If u(x,t) is the generalized solution of Eq. (1), then there are next* ^

*<sup>i</sup> nm* � *cui*, *<sup>t</sup>* � �

**Proof**. By the account of differentiation of regular generalized function rules

Here, *α*ð Þ *x*, *t δF*(*x*,*t*) is singular generalized function, which is a simple layer on

ð Þ¼ *ν*, *ν<sup>t</sup> gradF*, *F*, ð Þ*<sup>t</sup> = gradF*, *F*, k k ð Þ*<sup>t</sup> :*

If the right part of expression (7) is equal to zero, then the function *u*^(*x*,*t*) will satisfy to the Eq. (1) in a generalized sense. The natural requirement of the conti-

*ij u <sup>j</sup>*, *<sup>l</sup> and the velocity c of a wave front Ft coincides with one of ck*.

*<sup>i</sup> ν<sup>m</sup>* � *νtui*, *<sup>t</sup>* � �

*<sup>F</sup>νlδF*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � � � *ui* ½ �*FνtδF*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � �, *<sup>t</sup>* (7)

*<sup>α</sup>i*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>φ</sup>i*ð Þ *<sup>x</sup>*, *<sup>t</sup> dS x*ð Þ , *<sup>t</sup>* , <sup>∀</sup>*φ*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∈</sup> *DM <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � �

surface *F* is piecewise smooth with continuous normal on its smooth part.

named as *generalized solutions* of Eq. (1) (or *solutions in generalized sense)*.

*σm*

*Lij*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>u</sup>*^ *<sup>j</sup>*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>G</sup>*^*i*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>σ</sup><sup>m</sup>*

*F*

ð Þ *ν*1, … , *νN*, *ν<sup>t</sup>* is a unit vector, normal to characteristic surface *F*.

*dS x*ð Þ , *t* is the differential of the surface in a point ð Þ *x*, *t* and (*ν*,*νt*) =

*ij <sup>∂</sup><sup>m</sup> <sup>u</sup> <sup>j</sup>* � �

the surface *F* with specified density *α* ¼ ð Þ *α*1, … , *α<sup>M</sup>* :

If *F x*ð Þ¼ , *t* 0 is an equation of wave front, then

nuity of the solutions at transition through wave front *F*

<sup>þ</sup>*Cml*

<sup>ð</sup>*α*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>δ</sup>F*ð Þ *<sup>x</sup>*, *<sup>t</sup>* , *<sup>φ</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* Þ ¼ <sup>ð</sup>

ð Þ *<sup>x</sup>*, *<sup>τ</sup> <sup>φ</sup>i*ð Þ *<sup>x</sup>*, *<sup>τ</sup> dV x*ð Þ, <sup>∀</sup>*φ*<sup>∈</sup> *DM <sup>R</sup>N*þ<sup>1</sup> � �, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>M</sup>*

, continuous, twice differentiable

*<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � and its solutions in this space are

½ � *ui*ð Þ *x*, *t Ft* ¼ 0 (5)

*Ft* ¼ 0 (6)

*<sup>F</sup>δF*ð Þþ *x*, *t*

, and *n* is unit wave

function is presented in integral form:

*DOI: http://dx.doi.org/10.5772/intechopen.92449*

∞ð

*dτ* ð

*RN fi*

where *ν* is a normal vector to the characteristic surface *F* in *R<sup>N</sup>*þ<sup>1</sup>

�∞

Let *u x*ð Þ , *<sup>t</sup>* be the solution of Eq. (1) in *RN*þ<sup>1</sup>

Let us consider Eq. (1) in the space *D*<sup>0</sup>

*conditions on the jumps of its components and derivatives*:

^*f x*ð Þ , *<sup>t</sup>* , *<sup>φ</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � � <sup>¼</sup>

*generalized vector function*).

*D*0

**59**

*<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � �.

where *σ<sup>m</sup>*

[2], we receive:

*<sup>i</sup>* <sup>¼</sup> *<sup>C</sup>ml*

where *Gi* <sup>∈</sup>*L*<sup>2</sup> *<sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � and *<sup>δ</sup>ij* are Kronecker symbols; *<sup>∂</sup><sup>x</sup>* <sup>¼</sup> ð Þ *<sup>∂</sup>*1, … *<sup>∂</sup><sup>N</sup>* , *<sup>∂</sup><sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>=∂xi*, and *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>=∂<sup>t</sup>* are Partial derivatives; and also we will use following notations *ui*, *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> jui* and *ui*, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>tui*.

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

$$\mathcal{W}(n,v) = \mathcal{C}^{ml}\_{ij} n\_m n\_l v^i v^j > 0 \quad \forall n \neq 0, \quad v \neq 0$$

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)

$$\det\left\{\mathbf{C}\_{ij}^{ml}n\_m n\_l - c^2 \delta\_{ij}\right\} = \mathbf{0}, \left||n|| = \mathbf{1}\right.\tag{4}$$

has 2*M* valid roots (with the account of multiplicity):

$$\mathcal{L} = \pm c\_k(n) : \mathbf{0} < c\_k \le c\_{k+1}, k = \overline{1, M-1}$$

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 *D*<sup>0</sup> *<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � the space of generalized vector functions ^*f x*ð Þ¼ , *<sup>t</sup>* ^*<sup>f</sup>* <sup>1</sup>, … , ^*<sup>f</sup> <sup>M</sup>* � � determined on the space *DM <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � of finite and indefinitely *Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations… DOI: http://dx.doi.org/10.5772/intechopen.92449*

differentiable vector functions *<sup>φ</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>φ</sup>*<sup>1</sup> ð Þ , … , *<sup>φ</sup><sup>M</sup> :* For regular ^*f*, this linear function is presented in integral form:

$$\left(\widehat{f}\left(\mathbf{x},t\right),\rho(\mathbf{x},t)\right) = \int\_{-\infty}^{\infty} d\tau \int\_{\mathbb{R}^N} f\_i(\mathbf{x},\tau)\rho\_i(\mathbf{x},\tau)dV(\mathbf{x}), \quad \forall \rho \in D\_M\left(\mathbb{R}^{N+1}\right), \quad i = \overline{1,M}.$$

*dV* ¼ *dx*<sup>1</sup> … *dxN* (further, we shall say everywhere *generalized function* instead of *generalized vector function*).

Let *u x*ð Þ , *<sup>t</sup>* be the solution of Eq. (1) in *RN*þ<sup>1</sup> , continuous, twice differentiable almost everywhere, except for characteristic surface *F* which is motionless in *RN*þ<sup>1</sup> and mobile in *R<sup>N</sup>* (wave front *Ft*). On surface, *Ft* derivatives can have jumps. The equation of *F* is Eq. (4). We denote *ν* ¼ ð Þ¼ *n*1, … *nN*, *nt* ð Þ *n*, *tt* , *n* ¼ ð Þ *n*1, … *nN* , where *ν* is a normal vector to the characteristic surface *F* in *R<sup>N</sup>*þ<sup>1</sup> , and *n* is unit wave wave vector in *R<sup>N</sup>* directed in the direction of propagation *Ft*. 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*<sup>0</sup> *<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � 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 denote *u*^(*x*, *t*) = *u*(*x*, *t*), accordingly *G*^ (*x*,*t*) =*G*(*x*,*t*). Let *u*^(*x*,*t*) be the solution of Eq. (1) in *D*0 *<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � �.

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

$$[\mu\_i(\varkappa, t)]\_{F\_t} = \mathbf{0} \tag{5}$$

$$\left[\sigma\_i^m n\_m - \mathfrak{c} u\_i, \mathfrak{t}\_t\right]\_{F\_t} = \mathbf{0} \tag{6}$$

where *σ<sup>m</sup> <sup>i</sup>* <sup>¼</sup> *<sup>C</sup>ml ij u <sup>j</sup>*, *<sup>l</sup> and the velocity c of a wave front Ft coincides with one of ck*.

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

$$L\_{ij}(\partial\_{\mathbf{x}}, \partial\_{t})\hat{u}\_{j}(\mathbf{x}, t) + \hat{G}\_{i}(\mathbf{x}, t) = \left[\sigma\_{i}^{m}\nu\_{m} - \nu\_{t}u\_{i\ast\mathbf{t}}\right]\_{F}\delta\_{F}(\mathbf{x}, t) + $$

$$+ \mathcal{C}\_{ij}^{ml}\partial\_{m}\left(\left[u\_{j}\right]\_{F}\nu\_{l}\delta\_{F}(\mathbf{x}, t)\right) - \left(\left[u\_{i}\right]\_{F}\nu\_{l}\delta\_{F}(\mathbf{x}, t)\right)\_{t} \tag{7}$$

Here, *α*ð Þ *x*, *t δF*(*x*,*t*) is singular generalized function, which is a simple layer on the surface *F* with specified density *α* ¼ ð Þ *α*1, … , *α<sup>M</sup>* :

$$\rho\_i(a(\mathbf{x},t)\delta\_F(\mathbf{x},t),\rho(\mathbf{x},t)) = \int\_F a\_i(\mathbf{x},t)\rho\_i(\mathbf{x},t)dS(\mathbf{x},t), \forall \rho(\mathbf{x},t) \in D\_M(\mathbb{R}^{N+1})$$

*dS x*ð Þ , *t* is the differential of the surface in a point ð Þ *x*, *t* and (*ν*,*νt*) = ð Þ *ν*1, … , *νN*, *ν<sup>t</sup>* is a unit vector, normal to characteristic surface *F*.

If *F x*ð Þ¼ , *t* 0 is an equation of wave front, then

$$(\nu, \nu\_t) = (\mathsf{grad} F, F, \, \_t) / ||(\mathsf{grad} F, F, \, \_t)||.$$

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

generalized functions [1, 2] is used. At present, BIEM are applied very extensively

rated for boundary value problems of dynamics of elastic bodies [5–8].

*ij <sup>∂</sup>m∂<sup>l</sup>* � *<sup>δ</sup>ij∂*<sup>2</sup>

*Cml ij* <sup>¼</sup> *<sup>C</sup>lm*

Consider the second-order system of hyperbolic equations with constant

*ij* <sup>¼</sup> *Cml*

where *Gi* <sup>∈</sup>*L*<sup>2</sup> *<sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � and *<sup>δ</sup>ij* are Kronecker symbols; *<sup>∂</sup><sup>x</sup>* <sup>¼</sup> ð Þ *<sup>∂</sup>*1, … *<sup>∂</sup><sup>N</sup>* , *<sup>∂</sup><sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>=∂xi*,

and *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>=∂<sup>t</sup>* are Partial derivatives; and also we will use following notations

symmetry properties (3), satisfies the following condition of strict hyperbolicity:

*v j*

By the virtue of positive definiteness W, the characteristic equation of the

2 *δij*

*c* ¼ �*ck*ð Þ *n* : 0<*ck* ≤*ck*þ1, *k* ¼ 1, *M* � 1

*<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � the space of generalized vector functions

determined on the space *DM R<sup>N</sup>*þ<sup>1</sup> � � of finite and indefinitely

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 condi-

tions on jumps, it is convenient to use the theory of generalized functions.

Here everywhere like numbered indices indicate summation in specified limits

*ij nmnlvi*

*ij nmnl* � *c*

n o

*Lij*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>u</sup> <sup>j</sup>*ð Þþ *<sup>x</sup>*, *<sup>t</sup> Gi*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0, ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∈</sup>*R<sup>N</sup>*þ<sup>1</sup> (1)

*ji* <sup>¼</sup> *<sup>C</sup>ij*

*ij* , whose indices may be permitted in accordance with above indicated

>0 ∀*n* 6¼ 0, *v* 6¼ 0

*<sup>t</sup>* , *i*, *j* ¼ 1, *M*, *m*, *l* ¼ 1, *N* (2)

*ml* (3)

¼ 0, k k*n* ¼ 1 (4)

**2. Generalized solutions and conditions on wave fronts**

*Mathematical Theorems - Boundary Value Problems and Approximations*

*Lij*ð Þ¼ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>C</sup>ml*

*W n*ð Þ¼ , *<sup>v</sup> <sup>C</sup>ml*

*det Cml*

has 2*M* valid roots (with the account of multiplicity):

of their change (so as in tensor convolutions).

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 *<sup>R</sup>*<sup>3</sup> � *<sup>t</sup>*, 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 elabo-

to solve engineering problems.

coefficients:

system (1)

*ui*, *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> jui* and *ui*, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>tui*. The matrix *Cml*

Denote through *D*<sup>0</sup>

� �

^*f x*ð Þ¼ , *<sup>t</sup>* ^*<sup>f</sup>* <sup>1</sup>, … , ^*<sup>f</sup> <sup>M</sup>*

**58**

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$[\mu\_i(\mathbf{x}, t)]\_F = \mathbf{0} \tag{8}$$

*Ujk*ð Þ¼ *x*, 0 0 *for x* 6¼ 0 (13)

*<sup>M</sup> <sup>R</sup>N*þ<sup>1</sup> � �

ð Þ *ξ*,*ω* (14)

*Qjk*ð Þ *<sup>ξ</sup>*, *<sup>ω</sup>* , *<sup>Q</sup>*ð Þ¼ *λξ*, *λω <sup>λ</sup>*<sup>2</sup>*MQ*ð Þ *<sup>ξ</sup>*, *<sup>ω</sup>* (16)

ð*δikδ*ð Þ *x*, *t* , *φi*ð Þ *x*, *t* Þ ¼ *φk*ð Þ 0, 0 ∀*φ* ∈ *D*<sup>0</sup>

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations…*

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

*Ljk*ð Þ �*iξ*, �*iω Ukl*ð Þþ *ξ*,*ω δjl* ¼ 0, *j*, *k*, *l* ¼ 1, *M*

By permitting the system, we receive transformation of Green's matrix which by

Here, ð Þ¼ *<sup>ξ</sup>*, *<sup>ω</sup> <sup>ξ</sup>*1, … ,*ξN*,*<sup>ω</sup>* � � is the Fourier variables appropriate to ð Þ *<sup>x</sup>*, *<sup>t</sup>* .

*Ujk*ð Þ¼ *<sup>ξ</sup>*, *<sup>ω</sup> Qjk*ð Þ *<sup>ξ</sup>*, *<sup>ω</sup> <sup>Q</sup>*�<sup>1</sup>

where *Qjk* are the cofactors of the element with index (*k*, *j*) of the matrix

There are the following relations of symmetry and homogeneous:

By virtue of strong hyperbolicity characteristic equation,

*<sup>Q</sup>*ð Þ¼ *<sup>ξ</sup>*, *<sup>ω</sup>* det *Lkj*ð Þ �*iξ*, �*i<sup>ω</sup>* � �

*Qjk*ð Þ¼ *ξ*,*ω Qjk*ð Þ¼ �*ξ*, *ω Qjk*ð Þ *ξ*, �*ω* , *Q*ð Þ¼ *ξ*,*ω Q*ð Þ¼ �*ξ*, *ω Q*ð Þ *ξ*, �*ω* (15)

*Q*ð Þ¼ *ξ*, *ω* 0

**Theorem 3.1.** *If cq <sup>q</sup>* <sup>¼</sup> 1, *<sup>M</sup>* � � *are unitary roots of Eq. (4), then the Green's matrix*

� ð Þþ *<sup>e</sup>*, *<sup>x</sup> cq*ð Þ*<sup>e</sup> <sup>t</sup>* � *<sup>i</sup>*<sup>0</sup> � �<sup>1</sup>�*<sup>N</sup>* � ð Þ� *<sup>e</sup>*, *<sup>x</sup> cq*ð Þ*<sup>e</sup> <sup>t</sup>* � *<sup>i</sup>*<sup>0</sup> � �<sup>1</sup>�*<sup>N</sup>* n o*dS e*ð Þ

� �*=*2 *cqQmm e*,*cq*

� � � � , and *H t*ð Þ *is*

� � <sup>¼</sup> *Qjk <sup>e</sup>*,*cq*

**Theorem 3.2.** *If cq <sup>q</sup>* <sup>¼</sup> 1, *<sup>M</sup>* � � *are roots of Eq. (4) with multiplicity mq*, *then the*

has 2*M* roots. 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

virtue of differential polynomials uniformity looks like:

f g *L*ð Þ �*iξ*, �*iω* ; and *Q* is the symbol of operator *L*:

*Qjk*ð Þ¼ *λξ*, *λω <sup>λ</sup>*<sup>2</sup>*M*�<sup>2</sup>

been proved [10]:

*of system (1) has form*

*Heaviside's function.*

**61**

*Ujk*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>σ</sup>NH t*ð Þ<sup>X</sup>

*Green's matrix of system (1) has form*

*M*

ð

*Ajk e*,*cq* � �

k k¼*e* 1

*q*¼1

*where <sup>σ</sup><sup>N</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*π<sup>i</sup>* �*<sup>N</sup>*ð Þ *<sup>N</sup>* � <sup>2</sup> !, *Ajk <sup>e</sup>*,*cq*

Here, by definition,

*DOI: http://dx.doi.org/10.5772/intechopen.92449*

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

$$\left[\sigma\_i^m \nu\_m - \nu\_t \mu\_i,\_t\right]\_F = \mathbf{0} \tag{9}$$

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

$$\begin{aligned} [f(\mathfrak{x},t)]\_F &= \lim\_{\varepsilon \to +0} \left( f(\mathfrak{x} + \varepsilon \nu, t + \varepsilon \nu\_t) - f(\mathfrak{x} - \varepsilon \nu, t - \varepsilon \nu\_t) \right) \\ &= \lim\_{\varepsilon \to +0} \left( f(\mathfrak{x} + \varepsilon \nu, t) - f(\mathfrak{x} - \varepsilon \nu, t) \right) = [f(\mathfrak{x},t)]\_{F\_i}; \end{aligned}$$

therefore the condition (5) is equivalent to (8). If ð Þ *x*, *t* ∈*Ft*, then ð Þ *x* þ *cn*Δ*t*, *t* þ Δ*t* ∈*Ft*þΔ*<sup>t</sup>*. Therefore,

$$F(\mathbf{x} + c\mathbf{n}\Delta t, t + \Delta t) - F(\mathbf{x}, t) = \left(c(F, \, \_j n, \mathbf{n}\_j) + F, \_t\right)\Delta t = \mathbf{0}$$

From here, we have

$$\mathcal{c} = -F \mathfrak{z}\_t / \left( F \mathfrak{z}\_j , \mathfrak{n}\_j \right) = -\nu\_t / \sqrt{\nu\_i \nu\_i}$$

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

$$[n\_l u\_{i,t} + c u\_{i,l}]\_{F\_t} = 0, \quad i = \overline{1,M}, \quad l = \overline{1,N} \tag{10}$$

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*þ<sup>1</sup> , twice differentiable almost everywhere on *R<sup>N</sup>*þ<sup>1</sup> , and has limited number of piecewise smooth wave fronts on which conditions jumps (5) and (6) are carried out.
