**7. Singular boundary integral equations**

**Theorem 6.1.** *If u x*ð Þ , *t is a classical solution of the Dirichlet (Neumann) boundary value problem, then the generalized solution u can be represented as the the sum of the* ^

*<sup>i</sup>* ∗ *<sup>x</sup> u*<sup>1</sup>

*<sup>k</sup>*ð Þ *x H*�

*kj <sup>∂</sup>lV<sup>k</sup> i* � *<sup>x</sup> <sup>u</sup>*<sup>0</sup>

*<sup>D</sup>* ¼ *δ*ð Þ*t H*�

*<sup>k</sup>*ð Þ *x H*�

*<sup>S</sup>* ð Þþ *<sup>x</sup> <sup>∂</sup>tU*^ *<sup>k</sup>*

*<sup>i</sup>* � *<sup>x</sup> <sup>u</sup>*<sup>0</sup> *<sup>k</sup>* ð Þ *x H*�

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> <sup>j</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup> nm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>l</sup>:* (45)

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> jnm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>t</sup>* <sup>¼</sup>

*<sup>j</sup>*ð Þ *x nm*ð Þ *x δS*ð Þ *x*

¼

¼ *u*^*i*, *φ<sup>i</sup>* ð Þ*:*

*<sup>j</sup>*ð Þ *x nm*ð Þ *x δS*ð Þ *x δ*ð Þ*t*

*<sup>S</sup>* ð Þ *x* ,

*<sup>S</sup>* ð Þ *x δ*ð Þþ*t*

*kj <sup>u</sup> <sup>j</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup> nm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>l</sup>*

*<sup>S</sup>* ð Þþ*x*

¼

*<sup>S</sup>* ð Þþ*x*

*<sup>i</sup>* ∗ *gk*ð Þ *x*, *t δS*ð Þ *x H t*ð Þ� (44)

*<sup>j</sup>*ð Þ *x nm*ð Þ *x δS*ð Þ *x :*

*<sup>u</sup>*^*<sup>i</sup>* <sup>¼</sup> *<sup>U</sup><sup>k</sup>*

<sup>þ</sup>*∂tU<sup>k</sup> i* � *<sup>x</sup> <sup>u</sup>*<sup>0</sup> *<sup>k</sup>* ð Þ *x H*�

*∂ jH*�

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

*<sup>i</sup>* <sup>∗</sup> *<sup>G</sup>*^ *<sup>k</sup>* <sup>þ</sup> *<sup>U</sup>*^ *<sup>k</sup>*

differentiation rules for convolutions and generalized functions:

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> jnm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>l</sup>* <sup>¼</sup> *Cml*

*<sup>i</sup>* <sup>∗</sup> *gk*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *<sup>C</sup>ml*

*<sup>w</sup>*^*i*, *<sup>φ</sup><sup>i</sup>* ð Þ¼ *<sup>U</sup>*^ *<sup>k</sup>*

�

<sup>¼</sup> *Lkj*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>U</sup>*^ *<sup>k</sup>*

the solution of the problem is unique.

*gk*ð Þ *x*, *t δS*ð Þ *x H t*ð Þ *is a single layer on D.*

*<sup>S</sup>* ð Þ *<sup>x</sup>* \_

Eq. (1) in the form of the convolution

*<sup>w</sup>*^*i*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>k</sup>*

*<sup>i</sup>* <sup>∗</sup> *<sup>G</sup>*^ *<sup>k</sup>* <sup>þ</sup> *<sup>U</sup><sup>k</sup>*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> <sup>j</sup>*, *<sup>t</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup> nm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *<sup>C</sup>ml*

*<sup>D</sup>* ¼ �*<sup>n</sup> <sup>j</sup>δS*ð Þ *<sup>x</sup> H t*ð Þ, *<sup>∂</sup>tH*�

*Lkj*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>u</sup>*^ *<sup>j</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>G</sup>*^ *<sup>k</sup>*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>1</sup>

*<sup>δ</sup>*ð Þþ*<sup>t</sup> gk*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *<sup>C</sup>ml*

*<sup>i</sup>* � *x u*1 *<sup>k</sup>*ð Þ *x H*�

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> <sup>j</sup>*, *tnm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞþ *<sup>u</sup>*<sup>0</sup>

*<sup>i</sup>* <sup>∗</sup> *<sup>u</sup> <sup>j</sup>*, *tnm*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞþ *<sup>C</sup>ml*

Let us show that *<sup>w</sup>*^*i*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup>*^*i*ð Þ *<sup>x</sup>*, *<sup>t</sup>* . Indeed, <sup>∀</sup>*φ*<sup>∈</sup> *DN <sup>R</sup><sup>N</sup>*þ<sup>1</sup>

*<sup>i</sup>* <sup>∗</sup> *<sup>F</sup>*^*k*, *<sup>φ</sup><sup>i</sup>* 

*<sup>i</sup>* ∗ *u*^ *<sup>j</sup>*, *φ<sup>i</sup>*

*kj <sup>U</sup>*^ *<sup>k</sup>*

generalized functions, and taking into account the equalities

*<sup>S</sup>* ð Þþ *<sup>x</sup> <sup>U</sup><sup>k</sup>*

*Here, δ<sup>S</sup> is a singular generalized function that is a single layer on S (see [2]), and*

**Proof**. Applying the operator *Lij* to *u x* ^ð Þ , *t* , using the differentiation rules for

Next, we use the properties of Green's matrix to construct a weak solution of

The last convolution can be transformed using the relation (43) and applying the

<sup>¼</sup> *<sup>U</sup>*^ *<sup>k</sup>*

Here, *<sup>F</sup>*^*<sup>k</sup>* denotes the right-hand side of (44). Since *<sup>u</sup>*^*i*, *<sup>φ</sup><sup>i</sup>* ð Þ¼ 0, if *suppφ*<sup>∈</sup> *<sup>D</sup>*þ, it

<sup>¼</sup> *<sup>δ</sup> <sup>j</sup>*

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

*kj <sup>∂</sup>lV*^ *<sup>k</sup>*

*kj <sup>∂</sup>lV*^ *<sup>k</sup> <sup>i</sup>* � *<sup>x</sup> <sup>u</sup>*<sup>0</sup>

*<sup>i</sup> δ*ð Þ *x*, *t* ∗ *u*^ *<sup>j</sup>*, *φ<sup>i</sup>* 

*<sup>i</sup>* <sup>∗</sup> *Lkj*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>u</sup>*^ *<sup>j</sup>*, *<sup>φ</sup><sup>i</sup>* 

∈ *D*�. This implies the assertion of the theorem, since

*convolutions*

�*Cml kj <sup>∂</sup>lV<sup>k</sup>*

<sup>þ</sup>*u*<sup>0</sup> *<sup>k</sup>* ð Þ *x H*�

<sup>þ</sup>*U*^ *<sup>k</sup>*

*Cml kj <sup>∂</sup>tV*^ *<sup>k</sup>*

> <sup>¼</sup> *<sup>C</sup>ml kj <sup>∂</sup>lV*^ *<sup>k</sup>*

<sup>¼</sup> *<sup>C</sup>ml kj <sup>∂</sup>tV*^ *<sup>k</sup>*

follows that *w*^*i*ð Þ¼ *x*, *t* 0, *x*

of the problems.

**70**

**Lemma 7.1 (analogue of the Gauss formula).** *If S is an arbitrary closed Lyapunov surface in RN*, *then*

$$\int\_{\mathcal{S}} T\_k^{i(s)}(y-x,n(y))dS(y) = \delta\_k^i H\_{\mathcal{S}}^-(\infty)$$

*For x*∈ *S*, *the integral is singular and is understood in the sense of its principal value*. **Proof**. Convolution Eq. (27) with *H*� *<sup>S</sup>* ð Þ *x* and using the differentiation rules for convolutions yields

$$L\_{\vec{\eta}}(\partial\_{\mathbf{x}},\mathbf{0})U\_j^{k(\boldsymbol{s})} \* H\_S^{-}(\mathbf{x}) + \delta\_k^i H\_S^{-}(\mathbf{x}) = $$
 
$$0 = -C\_{\vec{\eta}}^{ml}U\_{jk}^{\boldsymbol{s}}, \boldsymbol{\iota} \* \boldsymbol{n}\_m \delta\_i^k H\_S^{-}(\mathbf{x}) = \int\_{\mathcal{S}} T\_k^{i(\boldsymbol{s})}(\boldsymbol{x} - \boldsymbol{y}, \boldsymbol{n}(\boldsymbol{y}))dS(\boldsymbol{y}) + \delta\_k^i H\_S^{-}(\mathbf{x}) = \mathbf{0}.$$

Using (29), we obtain the formula in the lemma. Since *Ti s*ð Þ *<sup>k</sup>* is regular for *x* ∉ *S*, the formula holds for such *x*. Let us prove the validity of this formula for boundary points.

Let *x*∈*S*. Define *Oε*ð Þ¼ *x* f g *y*∈*S* : ∥*y* � *x*∥< *ε* , *Sε*ð Þ¼ *x S* � *Oε*ð Þ *x* , Γ*ε*ð Þ¼ *x* f g *y* : ∥*y* � *x*∥ ¼ *ε* , Γ� *<sup>ε</sup>* ð Þ¼ *x* Γ*ε*ð Þ *x* ∩ *S*�, *and* Γ<sup>þ</sup> *<sup>ε</sup>* ð Þ¼ *x* Γ*ε*ð Þ *x* ∩ *S*þ.

Similarly, we obtain

$$\int\_{S\_{\varepsilon}} T\_k^{i(\boldsymbol{\nu})}(\boldsymbol{\y} - \boldsymbol{\varkappa}, n(\boldsymbol{\nu})) d\boldsymbol{S}(\boldsymbol{\wp}) + \int\_{\Gamma\_{\varepsilon}^{-}} T\_k^{i(\boldsymbol{\varkappa})}(\boldsymbol{\wp} - \boldsymbol{\varkappa}, n(\boldsymbol{\wp})) d\boldsymbol{S}(\boldsymbol{\wp}) = \mathbf{0}$$

$$\int\_{S\_{\varepsilon}} T\_k^{i(\boldsymbol{\varkappa})}(\boldsymbol{\wp} - \boldsymbol{\varkappa}, n(\boldsymbol{\wp})) d\boldsymbol{S}(\boldsymbol{\wp}) + \int\_{\Gamma\_{\varepsilon}^{+}} T\_k^{i(\boldsymbol{\varkappa})}(\boldsymbol{\wp} - \boldsymbol{\varkappa}, n(\boldsymbol{\wp})) d\boldsymbol{S}(\boldsymbol{\wp}) = \delta\_k^i$$

Since the outward normals to Γ� *<sup>ε</sup>* ð Þ *x* and Γ<sup>þ</sup> *<sup>ε</sup>* ð Þ *x* at opposite points *y*� and *y*<sup>þ</sup> of the sphere Γ*ε*ð Þ *x* coincide, i.e. *n y*� ð Þ¼ *x* � *y*� ð Þ*=ε* ¼ *y*ð Þ <sup>þ</sup> � *x =ε* ¼ *n y*<sup>þ</sup> ð Þ, while *<sup>y</sup>*ð Þ¼� <sup>þ</sup> � *<sup>x</sup> <sup>y</sup>*ð Þ � � *<sup>x</sup>* , we take into account the asymptotics of *<sup>T</sup>i s*ð Þ *<sup>k</sup>* and, according to Theorem 3.5, sum these two equalities and pass to the limit as *ε* ! 0, to obtain equality (30) for boundary points. The lemma is proved.

For *<sup>M</sup>* <sup>¼</sup> 1 and *<sup>L</sup>*1*<sup>j</sup>*ð Þ¼ *<sup>∂</sup>x*, 0 *<sup>∂</sup> <sup>j</sup><sup>∂</sup> <sup>j</sup>* <sup>¼</sup> *<sup>Δ</sup>*, 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

$$
\hat{u}\_k(\mathbf{x}, t) = \int\_D \left( T\_k^i(\mathbf{x} - \mathbf{y}, n(\mathbf{y}), t - \tau) u\_i(\mathbf{y}, t) + U\_k^i(\mathbf{x} - \mathbf{y}, t - \tau) \mathbf{g}\_i(\mathbf{y}, \tau) \right) d\mathcal{D}(\mathbf{y}, \tau) + \varepsilon
$$

$$
+ \int\_{S^-} \left( U\_{k \ast t}^i(\mathbf{x} - \mathbf{y}, t) u\_i^0(\mathbf{y}) + U\_k^i(\mathbf{x} - \mathbf{y}, t) u\_i^1(\mathbf{y}) \right) dV(\mathbf{y}) + U\_k^i \ast \hat{\mathbf{G}}\_i
$$

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 *T<sup>i</sup> <sup>k</sup>* has 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).

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

*S*

*S*�

Adding up and combining like terms, we derive the formula of the theorem for

**Theorem 7.2.** *The classical solution of the Dirichlet (Neumann) initial-boundary value problem for x*∈ *S and t*> 0 *satisfies the singular boundary integral equations* (*k* ¼ 1, *M*)

*dS y*ð Þ <sup>ð</sup>*<sup>t</sup>*

*<sup>k</sup>*ð Þ *x*, *t* � *<sup>x</sup> <sup>u</sup>*<sup>0</sup> *<sup>i</sup>* ð Þ*y H*� *<sup>S</sup>* ð Þ *x*

From these equations, we can determine the unknown boundary functions of the corresponding initial-boundary value problem. Next, the formulas of Theorem

*<sup>k</sup>*ð Þ *x*, *t* ∗ *gi*

0

*Wi d*ð Þ

� �, *<sup>t</sup>* <sup>þ</sup> *<sup>U</sup><sup>k</sup>*

The formula on the boundary yields boundary integral equations for solving

*S*

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

<sup>ð</sup>*y; <sup>τ</sup>*Þ þ *<sup>W</sup>i d*ð Þ

*Ui*

� �*dτ*<sup>þ</sup>

*<sup>k</sup> <sup>x</sup>*<sup>∗</sup> ð Þ � *<sup>y</sup>; <sup>t</sup> <sup>u</sup>*<sup>0</sup>

*Wi d*ð Þ

*<sup>k</sup> <sup>x</sup>*<sup>∗</sup> ð Þ � *<sup>y</sup>; n y*ð Þ*; <sup>t</sup> <sup>u</sup>*<sup>0</sup>

*<sup>i</sup>* ð Þ*<sup>y</sup>* � �, *tdV y*ð Þþ

*<sup>k</sup>* ¼

ð

*Ti s*ð Þ

ð Þ *x*, *t δs*ð Þ *x H t*ð Þ�

*<sup>k</sup>* ð Þ *x* � *y*, *n y*ð Þ, *t* � *τ ui*, *<sup>t</sup>*ð Þ *y*, *τ dτ*�

*<sup>i</sup>* � *x u*1 *<sup>k</sup>*ð Þ *x H*�

*<sup>S</sup>* ð Þ *x :*

*<sup>k</sup> <sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> ð Þ*dS y*ð Þþ

*k*

*S*

*<sup>k</sup> <sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> ð Þ*ui*ð Þ *<sup>y</sup>*, *<sup>t</sup> dS y*ð Þþ 0, 5*ui <sup>x</sup>*<sup>∗</sup> ð Þ , *<sup>t</sup> <sup>δ</sup><sup>i</sup>*

*<sup>k</sup>* <sup>ð</sup>*x*<sup>∗</sup> � *<sup>y</sup>; n y*ð Þ*; <sup>t</sup>* � *<sup>τ</sup>*Þ*ui*, *<sup>τ</sup>*ð*y; <sup>τ</sup>*<sup>Þ</sup>

*<sup>k</sup>* and

*<sup>i</sup>* ð Þ*y dS y*ð Þ�

Let *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>S</sup>*, *<sup>x</sup>*∈*S*� and *<sup>x</sup>* ! *<sup>x</sup>*<sup>∗</sup> . Then, since the convolutions containing *<sup>U</sup><sup>i</sup>*

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

*<sup>k</sup>* ð Þ *<sup>y</sup>* � *<sup>x</sup> ui*ð Þ *<sup>y</sup>; <sup>t</sup> dS y*ð Þþ <sup>ð</sup>

*<sup>k</sup> <sup>x</sup>*<sup>∗</sup> ð Þ � *<sup>y</sup>; <sup>t</sup>* � *<sup>τ</sup> gi*

*<sup>k</sup> <sup>x</sup>*<sup>∗</sup> ð Þ � *<sup>y</sup>; <sup>t</sup>* � *<sup>τ</sup> Gi*ð Þ *<sup>y</sup>; <sup>τ</sup> dV y*ð Þ*d<sup>τ</sup>*

*<sup>k</sup> <sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> ð Þ *ui*ð Þ� *<sup>y</sup>*, *<sup>t</sup> ui <sup>x</sup>*<sup>∗</sup> ð Þ ð Þ , *<sup>t</sup> dS y*ð Þþ *ui <sup>x</sup>*<sup>∗</sup> ð Þ , *<sup>t</sup> <sup>δ</sup><sup>i</sup>*

*<sup>k</sup> <sup>y</sup>* � *<sup>x</sup>*<sup>∗</sup> ð Þ*ui*ð Þ *<sup>y</sup>*, *<sup>t</sup> dS y*ð Þ� *ui <sup>x</sup>*<sup>∗</sup> ð Þ , *<sup>t</sup> <sup>V</sup>:P:*

*<sup>i</sup>*ð Þ*<sup>y</sup> dV y*ð Þþ <sup>ð</sup>

points as well.

*<sup>k</sup>* are continuous, we obtain

ð

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

*Ti s*ð Þ

*S*

*dS y*ð Þð*<sup>t</sup>*

0

*Ui*

*<sup>k</sup> <sup>x</sup>*<sup>∗</sup> ð Þ � *<sup>y</sup>; <sup>t</sup> <sup>u</sup>*<sup>1</sup>

lim*<sup>x</sup>*!*x*<sup>∗</sup> *uk*ð Þ¼ *<sup>x</sup>; <sup>t</sup> uk <sup>x</sup>*<sup>∗</sup> ð Þ¼ *; <sup>t</sup>*

� ð

þ ð

þ ð

ð

*Ti s*ð Þ

¼ *V:P:* ð

*S*

*Ti s*ð Þ

<sup>þ</sup>*ui <sup>x</sup>*<sup>∗</sup> ð Þ , *<sup>t</sup> <sup>δ</sup><sup>i</sup>*

initial-boundary value problems.

0, 5*uk*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>i</sup>*

*Ti s*ð Þ

*<sup>k</sup>* ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *n y*ð Þ, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

*S*

�*V:P:* ð

� ð

**73**

*S*

*Wi d*ð Þ

boundary points. The theorem is proved.

*<sup>k</sup>* ¼ *V:P:*

ð

*Ti s*ð Þ

*S*

*<sup>k</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *Gi*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>i</sup>*

*<sup>i</sup>* ð Þ*<sup>y</sup> dS y*ð Þþ *<sup>U</sup><sup>i</sup>*

*<sup>k</sup>* ð Þ *<sup>x</sup>* � *<sup>y</sup> ui*ð Þ *<sup>y</sup>*, *<sup>t</sup> dS y*ð Þ� <sup>ð</sup>

7.1 are used to determine the solution inside the domain.

*S*

*S*

*S*� *Ui*

*D*� *Ui*

<sup>¼</sup> lim*<sup>x</sup>*!*x*<sup>∗</sup>

*Wi d*ð Þ

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

$$\begin{split} \dot{u}\_{k} &= U^{i}\_{k}(\boldsymbol{\kappa},t) \ast \mathbf{G}\_{i}(\boldsymbol{\kappa},t) + U^{i}\_{k}(\boldsymbol{\kappa},t) \ast \mathbf{g}\_{i}(\boldsymbol{\kappa},t) \delta\_{i}(\boldsymbol{\kappa})H(t) - \\ &- \int\_{\mathcal{S}} T^{i(\boldsymbol{\varepsilon})}\_{k}(\boldsymbol{\kappa}-\boldsymbol{\jmath}) u\_{i}(\boldsymbol{\jmath},t) d\boldsymbol{S}(\boldsymbol{\jmath}) - \int\_{\mathcal{S}} d\boldsymbol{\mathcal{S}}(\boldsymbol{\jmath}) \int\_{0}^{t} W^{i(d)}\_{k}(\boldsymbol{\kappa}-\boldsymbol{\jmath},n(\boldsymbol{\jmath}),t-\tau) u\_{i}, \boldsymbol{\iota}\_{i}(\boldsymbol{\jmath},\tau) d\tau - \\ &- \int\_{\mathcal{S}} W^{i(d)}\_{k}(\boldsymbol{\kappa}-\boldsymbol{\jmath},n(\boldsymbol{\jmath}),t) u^{0}\_{i}(\boldsymbol{\nu}) d\boldsymbol{S}(\boldsymbol{\wp}) + \left(U^{i}\_{k}(\boldsymbol{\kappa},t) \quad \ast\_{\boldsymbol{\varkappa}} u^{0}\_{i}(\boldsymbol{\varmu}) H^{-}\_{\boldsymbol{\kappa}}(\boldsymbol{\kappa})\right)\_{t} \end{split}$$

*For x*∈ *S, 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

$$\begin{split} \hat{u}\_{k} &= \int\_{\mathcal{S}} dS(\mathbf{y}) \int\_{0}^{t} (U\_{k}^{i}(\mathbf{x}-\mathbf{y},t-\tau)\mathbf{g}\_{i}(\mathbf{y},\tau) - W\_{k}^{i}(\mathbf{x}-\mathbf{y},n(\mathbf{y}),t-\tau)\mathbf{u}\_{i},\tau(\mathbf{y},\tau))d\tau - \\ &- \int\_{\mathcal{S}} W\_{k}^{i}(\mathbf{x}-\mathbf{y},n(\mathbf{y}),t)u\_{i}^{0}(\mathbf{y})dS(\mathbf{y}) + \partial\_{t} \int\_{\mathcal{S}} U\_{k}^{i}(\mathbf{x}-\mathbf{y},t)u\_{i}^{0}(\mathbf{y})dS^{-}(\mathbf{y}) + \\ &+ \int\_{\mathcal{S}} U\_{k}^{i}(\mathbf{x}-\mathbf{y},t)u\_{i}^{1}(\mathbf{y})dV(\mathbf{y}) + \int\_{D^{-}} U\_{k}^{i}(\mathbf{x}-\mathbf{y},t-\tau)G\_{i}(\mathbf{y},\tau)dV(\mathbf{y})d\tau \end{split}$$

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

**Remark.** If *N* is odd, then, since *U* is singular, the integrals involving *U* are 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

$$\begin{aligned} \int\_{\mathcal{S}} dS(\boldsymbol{\mathcal{y}}) \int\_{0}^{t} W\_{k}^{i}(\boldsymbol{\mathcal{x}} - \boldsymbol{\mathcal{y}}, n(\boldsymbol{\mathcal{y}}), t - \tau) d\_{\tau} u\_{i}(\boldsymbol{\mathcal{y}}, \tau) &= \\ \int\_{\mathcal{S}} T\_{k}^{i(\boldsymbol{\mathcal{x}})} (\boldsymbol{\mathcal{x}} - \boldsymbol{\mathcal{y}}) \Big( u\_{i}(\boldsymbol{\mathcal{y}}, t) - u\_{i}^{0}(\boldsymbol{\mathcal{y}}) \Big) dS(\boldsymbol{\mathcal{y}}) \\ + \int\_{\mathcal{S}} dS(\boldsymbol{\mathcal{y}}) \int\_{0}^{t} W\_{k}^{i(\boldsymbol{d})} (\boldsymbol{\mathcal{x}} - \boldsymbol{\mathcal{y}}, n(\boldsymbol{\mathcal{y}}), t - \tau) u\_{i}, \tau(\boldsymbol{\mathcal{y}}, \tau) d\tau \end{aligned}$$

Here, the first integral is singular for *x*∈*S* and 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.

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

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

Let *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>S</sup>*, *<sup>x</sup>*∈*S*� and *<sup>x</sup>* ! *<sup>x</sup>*<sup>∗</sup> . Then, since the convolutions containing *<sup>U</sup><sup>i</sup> <sup>k</sup>* and *Wi d*ð Þ *<sup>k</sup>* are continuous, we obtain

$$\begin{split} \lim\_{\mathbf{x}\to\mathbf{x}^{\*}} u\_{k}(\mathbf{x},t) &= u\_{k}(\mathbf{x}^{\*},t) = \\ &= \lim\_{\mathbf{x}\to\mathbf{x}^{\*}} \Big[ T\_{k}^{(\mathbf{x})}(\mathbf{y}-\mathbf{x})u\_{i}(\mathbf{y},t)dS(\mathbf{y}) + \int\_{\mathbf{S}} W\_{k}^{i(d)}(\mathbf{x}^{\*}-\mathbf{y},n(\mathbf{y}),t)u\_{i}^{0}(\mathbf{y})dS(\mathbf{y}) - \\ & - \int\_{\mathbf{S}} dS(\mathbf{y}) \Big] \Big( U\_{k}^{i}(\mathbf{x}^{\*}-\mathbf{y},t-\tau)g\_{i}(\mathbf{y},\tau) + W\_{k}^{i(d)}(\mathbf{x}^{\*}-\mathbf{y},n(\mathbf{y}),t-\tau)u\_{i},\tau(\mathbf{y},\tau) \Big) d\tau + \\ & + \int\_{\mathbf{S}} U\_{k}^{i}(\mathbf{x}^{\*}-\mathbf{y},t)u\_{i}^{1}(\mathbf{y})dV(\mathbf{y}) + \int\_{\mathbf{S}} \Big( U\_{k}^{i}(\mathbf{x}^{\*}-\mathbf{y},t)u\_{i}^{0}(\mathbf{y}) \Big), dW(\mathbf{y}) + \\ & + \int\_{\mathbf{D}} U\_{k}^{i}(\mathbf{x}^{\*}-\mathbf{y},t-\tau)G\_{i}(\mathbf{y},\tau)dV(\mathbf{y})d\tau \\ \end{split}$$

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

$$\begin{aligned} &\int\_{\mathcal{S}} T\_k^{i(\boldsymbol{\varrho})} (\boldsymbol{y} - \boldsymbol{\varkappa}^\*) (\boldsymbol{u}\_i(\boldsymbol{y}, t) - \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t)) dS(\boldsymbol{y}) + \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t) \delta\_k^i = \\ &= V.P.\Big[\int\_{\mathcal{S}} T\_k^{i(\boldsymbol{\varrho})} (\boldsymbol{y} - \boldsymbol{\varkappa}^\*) \boldsymbol{u}\_i(\boldsymbol{y}, t) dS(\boldsymbol{y}) - \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t) V.P.\Big]\_{\mathcal{S}}^{i(\boldsymbol{\varkappa})} (\boldsymbol{y} - \boldsymbol{\varkappa}^\*) dS(\boldsymbol{y}) + \int\_{\mathcal{S}} \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t) \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t) dS(\boldsymbol{y}) \\ &+ \boldsymbol{u}\_i(\boldsymbol{\varkappa}^\*, t) \delta\_k^i = V.P.\Big]\_{\mathcal{S}}^{i(\boldsymbol{\varkappa})} (\boldsymbol{y} - \boldsymbol{\varkappa}^\*) \boldsymbol{u}\_i(\boldsymbol{y}, t) dS(\boldsymbol{y}) + \mathbf{0}, \mathsf{5}u\_i(\boldsymbol{\varkappa}^\*, t) \delta\_k^i \Big] \end{aligned}$$

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 x*∈ *S and t*> 0 *satisfies the singular boundary integral equations* (*k* ¼ 1, *M*)

$$\begin{aligned} 0, \mathfrak{H}u\_k(\mathbf{x}, t) &= U\_k^i(\mathbf{x}, t) \ast G\_i(\mathbf{x}, t) + U\_k^i(\mathbf{x}, t) \ast g\_i(\mathbf{x}, t) \delta\_i(\mathbf{x}) H(t) - \\ &- V.P.\left[\int\_S^{i(t)} (\mathbf{x} - \mathbf{y}) u\_i(\mathbf{y}, t) d\mathbf{S}(\mathbf{y}) - \int\_S^t d\mathbf{S}(\mathbf{y}) \int\_0^t W\_k^{i(d)}(\mathbf{x} - \mathbf{y}, n(\mathbf{y}), t - \tau) u\_{i, t}(\mathbf{y}, \tau) d\tau - \right. \\ &- \left[\int\_S^{i(d)} (\mathbf{x} - \mathbf{y}, n(\mathbf{y}), t) u\_i^0(\mathbf{y}) d\mathbf{S}(\mathbf{y}) + \left(U\_k^i(\mathbf{x}, t) \underset{\mathbf{x}}{\ast} u\_i^0(\mathbf{y}) H\_S^-(\mathbf{x})\right)\_{\mathbb{H}} + U\_i^k \underset{\mathbf{x}}{\ast} u\_k^1(\mathbf{x}) H\_S^-(\mathbf{x}). \end{aligned}$$

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.

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 *T<sup>i</sup>*

strong singularities on the fronts. However, the primitives of the matrix introduced in

ð Þ *x*, *t δs*ð Þ *x H t*ð Þ�

*Wi d*ð Þ

*<sup>k</sup>*ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *n y*ð Þ, *<sup>t</sup>* � *<sup>τ</sup> ui*, *<sup>τ</sup>*ð Þ *<sup>y</sup>*, *<sup>τ</sup>* � �*dτ*�

ð

*S*� *Ui*

*<sup>k</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *<sup>x</sup> <sup>u</sup>*<sup>0</sup>

*<sup>k</sup>*ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

*<sup>k</sup>*ð Þ *x* � *y*, *t* � *τ Gi*ð Þ *y*, *τ dV y*ð Þ*dτ*

*<sup>S</sup>* ð Þ *<sup>x</sup>* � �, *<sup>t</sup>*

*<sup>k</sup>* ð Þ *x* � *y*, *n y*ð Þ, *t* � *τ ui*, *<sup>t</sup>*ð Þ *y*, *τ dτ*�

*<sup>i</sup>* ð Þ*y H*�

*<sup>i</sup>* ð Þ*y dS*�ð Þþ*y*

Section 3 can be used to construct integral representations of formula (44). **Theorem 7.1.** *If u is a classical solution of the boundary value problem, then*

ð

*dS y*ð Þ ð*t*

*<sup>i</sup>* ð Þ*<sup>y</sup> dS y*ð Þþ *<sup>U</sup><sup>i</sup>*

0

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

ð Þ� *<sup>y</sup>*, *<sup>τ</sup> <sup>W</sup><sup>i</sup>*

*<sup>i</sup>* ð Þ*<sup>y</sup> dS y*ð Þþ *<sup>∂</sup><sup>t</sup>*

ð

*D*� *Ui*

Here, all the integrals are regular for interior points and singular for boundary points. **Remark.** If *N* is odd, then, since *U* is singular, the integrals involving *U* are 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

It is easy to see that, for zero initial data, the last three integrals (in the convo-

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

Here, the first integral is singular for *x*∈*S* and exists in the sense of its principal value by Lemma 7.1, while the second integral is regular. Then for interior points,

*<sup>k</sup>* ð Þ *<sup>x</sup>* � *<sup>y</sup> ui*ð Þ� *<sup>y</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

*<sup>k</sup>*ð Þ *x* � *y*, *n y*ð Þ, *t* � *τ dτui*ð Þ¼ *y*, *τ*

*<sup>i</sup>* ð Þ*<sup>y</sup>* � �*dS y*ð Þ

*<sup>k</sup>* ð Þ *x* � *y*, *n y*ð Þ, *t* � *τ ui*, *<sup>τ</sup>*ð Þ *y*, *τ dτ*

*<sup>i</sup>*ð Þ*y dV y*ð Þþ

*S*

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

*<sup>k</sup>*ð Þ *x*, *t* ∗ *gi*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*<sup>u</sup>*^*<sup>k</sup>* <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

� ð

� ð

*u*^*<sup>k</sup>* ¼ ð

lution) vanish.

**72**

*S*

*S*

*S*

� ð

> þ ð

*S*� *Ui*

*S W<sup>i</sup>*

*dS y*ð Þ ð*t*

0

*Ui*

*<sup>k</sup>*ð Þ *x* � *y*, *t* � *τ gi*

*<sup>k</sup>*ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *n y*ð Þ, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

*<sup>k</sup>*ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>1</sup>

representations were constructed in [4].

ð

*dS y*ð Þ ð*t*

> ¼ ð

þ ð

we obtain the formula of the theorem.

*S*

*S*

0 *W<sup>i</sup>*

*Ti s*ð Þ

*dS y*ð Þ ð*t*

0

*Wi d*ð Þ

*S*

*Ti s*ð Þ

*Wi d*ð Þ

*<sup>k</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *Gi*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>i</sup>*

*<sup>k</sup>* ð Þ *x* � *y ui*ð Þ *y*, *t dS y*ð Þ�

*<sup>k</sup>* ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *n y*ð Þ, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

*<sup>k</sup>* has
