**4. Statement of the initial BVP**

Consider the system of strict hyperbolic equations (1). Assume that *x*∈*S*� ⊂*R<sup>N</sup>*, where *S*� is an open bounded set; ð Þ *x*, *t* ∈ *D*�, *D*� ¼ *S*� � ð Þ 0, ∞ , *D*� *<sup>t</sup>* ¼ *S*� � ð Þ 0, *t* , *t*>0; *D* ¼ *S* � ð Þ 0, ∞ , and *Dt* ¼ *S* � ð Þ 0, *t* .

The boundary *S* of *S*� is a Lyapunov surface with a continuous outward normal *n x*ð Þ ð Þ ∥*n*∥ ¼ 1 :

$$\|\|n(\varkappa\_2) - n(\varkappa\_1)\|\|\ = O\left(\|\varkappa\_2 - \varkappa\_1\|^\beta\right), \ \beta > 0, \ \varkappa\_1 \in \mathbb{S}, \ \varkappa\_2 \in \mathbb{S}.$$

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

$$G \to 0 \quad \text{as} \quad t \to +\infty, \quad \forall \mathfrak{x} \in \mathbb{S}^-.$$

Furthermore, *u*∈*C D*� ð Þ þ *D* , where *u* is a twice differentiable vector function almost everywhere on *D*�, except for possibly the characteristic surfaces (*F*) in *R<sup>N</sup>*þ<sup>1</sup> , which correspond to the moving wavefronts (*Ft*) *R<sup>N</sup>*. 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* � 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*

$$u\_i(\mathbf{x}, \mathbf{0}) = u\_i^0(\mathbf{x}), \quad \mathbf{x} \in \mathbb{S}^- + \mathbb{S} \tag{33}$$

$$u\_{i,t}(\mathbf{x}, \mathbf{0}) = u\_i^1(\mathbf{x}), \quad \mathbf{x} \in \mathbb{S}^-; \tag{34}$$

*the Dirichlet conditions*

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations… DOI: http://dx.doi.org/10.5772/intechopen.92449*

$$u\_i(\mathbf{x}, t) = u\_i^S(\mathbf{x}, t), \quad \mathbf{x} \in \mathbb{S}, \quad t \ge 0; \tag{35}$$

*and the Neumann-type conditions*

**Theorem 3.5.** *The following representations take place*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup>k s*ð Þ

*Mathematical Theorems - Boundary Value Problems and Approximations*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>T</sup>k s*ð Þ

*ik*ð Þ *ex* , *<sup>T</sup>k s*ð Þ

*ik*ð Þ *ex* , *<sup>T</sup>k s*ð Þ

*<sup>i</sup>* ð Þ *<sup>x</sup> H t*ðÞþ *<sup>V</sup>k d*ð Þ

*<sup>i</sup>* ð Þ *<sup>x</sup> H t*ðÞþ *<sup>W</sup>k d*ð Þ

*<sup>i</sup>* ð Þ� *<sup>x</sup>* <sup>∥</sup>*x*∥�<sup>1</sup>

*<sup>i</sup>* ð Þ� *<sup>x</sup>* <sup>∥</sup>*x*∥�*N*þ<sup>1</sup>

*<sup>i</sup>* ð Þ¼ *x*, *t* 0 *for* ∥*x*∥> max

Consider the system of strict hyperbolic equations (1). Assume that *x*∈*S*� ⊂*R<sup>N</sup>*,

The boundary *S* of *S*� is a Lyapunov surface with a continuous outward normal

<sup>∥</sup>*n x*ð Þ� <sup>2</sup> *n x*ð Þ<sup>1</sup> <sup>∥</sup> <sup>¼</sup> *<sup>O</sup>* <sup>∥</sup>*x*<sup>2</sup> � *<sup>x</sup>*1∥*<sup>β</sup>* , *<sup>β</sup>* <sup>&</sup>gt;0, *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> *<sup>S</sup>*, *<sup>x</sup>*<sup>2</sup> <sup>∈</sup>*S:*

*G* ! 0 as *t* ! þ∞, ∀*x*∈*S*�*:*

Furthermore, *u*∈*C D*� ð Þ þ *D* , where *u* is a twice differentiable vector function almost everywhere on *D*�, except for possibly the characteristic surfaces (*F*) in

It is assumed that the number of wavefronts is finite and each front is almost

**Problem 1.** Find a solution of system (1) satisfying conditions (5)–(7) if the

, which correspond to the moving wavefronts (*Ft*) *R<sup>N</sup>*. On them, conditions (5)

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

where *S*� is an open bounded set; ð Þ *x*, *t* ∈ *D*�, *D*� ¼ *S*� � ð Þ 0, ∞ , *D*�

*<sup>i</sup>* ð Þ *x*, *t* , (30)

*<sup>i</sup>* ð Þ *x*, *t* , (31)

*ik*ð Þ *ex* , *N* ¼ 2,

max ∥*e*∥¼1

*ck*ð Þ*e t:*.

*<sup>i</sup>* ð Þ *x* , *x*∈ *S*� þ *S* (33)

*<sup>i</sup>*ð Þ *x* , *x*∈ *S*�; (34)

*ik*ð Þ *ex* , *N* >2*:* (32)

*ck*ð Þ*e t*,

*<sup>t</sup>* ¼ *S*� �

*<sup>i</sup>* ð Þ *x H t*ð Þ are regular functions for *x* 6¼ 0. As ∥*x*∥ ! 0,

*BN*

*BN*

*ik*ð Þ*e are continuous and bounded functions on the*

*k*¼1, *M*

min ∥*e*∥¼1

*k*¼1, *M*

*<sup>i</sup> are regular functions that are continuous at x* ¼

*V*^ *k*

*<sup>W</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* ð Þ *<sup>x</sup> H t*ð Þ and *<sup>T</sup>k s*ð Þ

*<sup>i</sup>* ð Þ� *<sup>x</sup>* ln <sup>∥</sup>*x*∥*AN*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>0</sup> *<sup>W</sup>k d*ð Þ

ð Þ 0, *t* , *t*>0; *D* ¼ *S* � ð Þ 0, ∞ , and *Dt* ¼ *S* � ð Þ 0, *t* .

everywhere a Lyapunov surface of dimension *N* � 1.

boundary values of the following functions are given:

*ui*ð Þ¼ *<sup>x</sup>*, 0 *<sup>u</sup>*<sup>0</sup>

*ui*, *<sup>t</sup>*ð Þ¼ *<sup>x</sup>*, 0 *<sup>u</sup>*<sup>1</sup>

*AN*

*ik*ð Þ*<sup>e</sup>* , *and B<sup>N</sup>*

*<sup>i</sup> andWk d*ð Þ

and for odd *N*, these relations hold for ∥*x*∥< min

where *Uk s*ð Þ

*Uk s*ð Þ

*n x*ð Þ ð Þ ∥*n*∥ ¼ 1 :

*R<sup>N</sup>*þ<sup>1</sup>

**66**

and (6) are satisfied.

*the initial values*

*the Dirichlet conditions*

*Uk s*ð Þ

*Here, ex* <sup>¼</sup> *<sup>x</sup>=*∥*x*∥, *AN*

*Vk d*ð Þ

**4. Statement of the initial BVP**

*sphere* <sup>∥</sup>*e*<sup>∥</sup> <sup>¼</sup> <sup>1</sup>*, and Vk d*ð Þ

0*and t*> 0*. For any N,*

*<sup>i</sup>* ð Þ� *<sup>x</sup>* <sup>∥</sup>*x*∥�*N*þ<sup>2</sup>

$$
\sigma\_i^l(\mathbf{x}, t) n\_l(\mathbf{x}) = \mathbf{g}\_i(\mathbf{x}, t), \quad \mathbf{x} \in \mathbb{S}, \quad t \ge 0, \quad i = \overline{1, N}. \tag{36}
$$

**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

$$
\mu\_i(\mathfrak{x}, \mathbf{0}) = \mathfrak{u}\_i^0(\mathfrak{x}), \quad \mathfrak{u}\_i^{\mathrm{S}},\_t(\mathfrak{x}, \mathbf{0}) = \mathfrak{u}\_i^1(\mathfrak{x}), \quad \mathfrak{x} \in \mathfrak{S}.
$$

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.
