**6. 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*�. The characteristic function *H*� *<sup>S</sup>* ð Þ *x* of a set *S*� is defined for *x*∈*S* as

$$H\_{\mathbb{S}}^{-}(\mathfrak{x}) = \mathbf{1}/\mathfrak{2} \tag{38}$$

The Heaviside function *H t*ð Þ is extended to zero by setting *H*ð Þ¼ 0 1*=*2. Define the characteristic function of *D*� as

$$H\_D^{-}(\mathbf{x}, t) = H\_S^{-}(\mathbf{x}) H(t) \tag{39}$$

Accordingly, for *u* defined on *D*�, we introduce the generalized function

$$
\hat{u}(\mathbf{x},t) = \mathfrak{u}H\_D^-(\mathbf{x},t),\tag{40}
$$

which is defined on the entire space *R<sup>N</sup>*þ<sup>1</sup> . Similarly,

$$
\hat{G}\_k(\mathbf{x}, t) = G\_k H\_D^-(\mathbf{x}, t). \tag{41}
$$

Let *<sup>U</sup>*^ *<sup>k</sup> <sup>i</sup>* ð Þ *x*, *t* denotes the Green's matrix, i.e. the fundamental solution of Eq. (1) that corresponds to the function *Fi* <sup>¼</sup> *<sup>δ</sup><sup>k</sup> <sup>i</sup> δ*ð Þ *x δ*ð Þ*t* and satisfies the conditions

$$
\hat{U}\_i^k(\mathbf{x}, \mathbf{0}) = \mathbf{0}, \quad \hat{U}\_i^k \llcorner (\mathbf{x}, \mathbf{0}) = \mathbf{0}, \quad \mathbf{x} \neq \mathbf{0} \tag{42}
$$

For system (1), such a matrix was constructed in [10]. The primitive of Green's matrix with respect to *t* is defined as

$$
\hat{\boldsymbol{V}}\_i^k(\mathbf{x}, t) = \hat{\boldsymbol{U}}\_i^k(\mathbf{x}, t) \, \ast \boldsymbol{H}(t) \quad \Rightarrow \quad \partial\_t \hat{\boldsymbol{V}}\_i^k = \hat{\boldsymbol{U}}\_i^k. \tag{43}
$$

Here and below, the star denotes the complete convolution with respect to ð Þ *x*, *t* , 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 *<sup>t</sup>*. Clearly, the convolution is the solution of Eq. (1) at *Fi* <sup>¼</sup> *<sup>δ</sup><sup>k</sup> <sup>i</sup> δ*ð Þ *x H t*ð Þ.

**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* ^ *convolutions*

$$\hat{u}\_{i} = U\_{i}^{k} \* \hat{G}\_{k} + U\_{i}^{k} \quad \* \ \boldsymbol{u}\_{\cdot}^{1}(\boldsymbol{\kappa})H\_{\mathcal{S}}^{-}(\boldsymbol{\kappa}) +$$

$$+ \partial\_{t} U\_{i}^{k} \* \boldsymbol{u}\_{\cdot}^{0}(\boldsymbol{\kappa})H\_{\mathcal{S}}^{-}(\boldsymbol{\kappa}) + U\_{i}^{k} \* \boldsymbol{g}\_{k}(\boldsymbol{\kappa}, t)\delta\_{\mathcal{S}}(\boldsymbol{\kappa})H(t) - \tag{44}$$

$$- \mathcal{L}\_{kj}^{ml} \,\partial\_{l} V\_{i}^{k} \* \boldsymbol{u}\_{j\cdot\cdot t}(\boldsymbol{\kappa}, t)\boldsymbol{n}\_{m}(\boldsymbol{\kappa})\delta\_{\mathcal{S}}(\boldsymbol{\kappa})H(t) - \mathcal{L}\_{kj}^{ml} \,\partial\_{l} V\_{i}^{k} \* \boldsymbol{u}\_{j}^{0}(\boldsymbol{\kappa})\boldsymbol{n}\_{m}(\boldsymbol{\kappa})\delta\_{\mathcal{S}}(\boldsymbol{\kappa}).$$

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,

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

**Lemma 7.1 (analogue of the Gauss formula).** *If S is an arbitrary closed*

*<sup>k</sup>* ð Þ *<sup>y</sup>* � *<sup>x</sup>*, *n y*ð Þ *dS y*ð Þ¼ *<sup>δ</sup><sup>i</sup>*

*<sup>j</sup>* ∗ *H*�

*Ti s*ð Þ

the formula holds for such *x*. Let us prove the validity of this formula for boundary

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*

Γ� *ε*

Γþ *ε*

*<sup>ε</sup>* ð Þ *x* and Γ<sup>þ</sup>

to Theorem 3.5, sum these two equalities and pass to the limit as *ε* ! 0, to obtain

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

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

ð Þ *<sup>y</sup>*, *<sup>τ</sup>* � �*dD y*ð Þþ , *<sup>τ</sup>*

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

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

formula for the double-layer potential of Laplace equation (see [2]).

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

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

*Ti s*ð Þ

*Ti s*ð Þ

ð

*S*

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

*<sup>S</sup>* ð Þþ *<sup>x</sup> <sup>δ</sup><sup>i</sup>*

*kH*� *<sup>S</sup>* ð Þ *x*

*kH*� *<sup>S</sup>* ð Þ¼ *x*

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

*<sup>ε</sup>* ð Þ¼ *x* Γ*ε*ð Þ *x* ∩ *S*þ.

*<sup>k</sup>* ð Þ *y* � *x*, *n y*ð Þ *dS y*ð Þ¼ 0

*<sup>k</sup>* ð Þ *<sup>y</sup>* � *<sup>x</sup>*, *n y*ð Þ *dS y*ð Þ¼ *<sup>δ</sup><sup>i</sup>*

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

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

*<sup>S</sup>* ð Þ *x* and using the differentiation rules for

*kH*�

*<sup>S</sup>* ð Þ¼ *x* 0

*<sup>k</sup>* is regular for *x* ∉ *S*,

*k*

*<sup>k</sup>* and, according

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

*<sup>ε</sup>* ð Þ *x* at opposite points *y*� and *y*<sup>þ</sup> of the

we examine the properties of the functional matrices involved.

**7. Singular boundary integral equations**

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

ð

*Ti s*ð Þ

*Lij*ð Þ *<sup>∂</sup>x*, 0 *<sup>U</sup>k s*ð Þ

Using (29), we obtain the formula in the lemma. Since *Ti s*ð Þ

*<sup>ε</sup>* ð Þ¼ *x* Γ*ε*ð Þ *x* ∩ *S*�, *and* Γ<sup>þ</sup>

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

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

equality (30) for boundary points. The lemma is proved.

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

*<sup>i</sup> H*� *<sup>S</sup>* ð Þ¼ *x*

*S*

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

*jk*, *<sup>l</sup>* ∗ *nmδ<sup>k</sup>*

*Lyapunov surface in RN*, *then*

convolutions yields

points.

¼ �*Cml ij U<sup>s</sup>*

f g *y* : ∥*y* � *x*∥ ¼ *ε* , Γ�

Similarly, we obtain

ð

*Ti s*ð Þ

*Ti s*ð Þ

Since the outward normals to Γ�

*Sε*

ð

*Sε*

ð

*Ti*

�

*Ui*

*D*

þ ð

*S*�

*u*^*k*ð Þ¼ *x*, *t*

**71**

*Here, δ<sup>S</sup> is a singular generalized function that is a single layer on S (see [2]), and gk*ð Þ *x*, *t δS*ð Þ *x H t*ð Þ *is a single layer on D.*

**Proof**. Applying the operator *Lij* to *u x* ^ð Þ , *t* , using the differentiation rules for generalized functions, and taking into account the equalities

$$
\partial\_j H^-\_D = -n\_j \delta\_\mathbb{S}(\mathfrak{x}) H(t), \\
\partial\_t H^-\_D = \delta(t) H^-\_\mathbb{S}(\mathfrak{x}),
$$

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

$$\begin{aligned} L\_{\dot{\mathbb{K}}}(\partial\_{\mathbf{x}},\partial\_{t})\hat{u}\_{j}(\mathbf{x},t) &= \hat{G}\_{k}(\mathbf{x},t) + \boldsymbol{u}\_{k}^{1}(\mathbf{x})H\_{\mathbb{S}}^{-}(\mathbf{x})\delta(t) + \\ + \boldsymbol{u}\_{k}^{0}(\mathbf{x})H\_{\mathbb{S}}^{-}(\mathbf{x})\dot{\delta}(t) + \mathbf{g}\_{k}(\mathbf{x},t)\delta\_{\mathbb{S}}(\mathbf{x})H(t) - \mathbf{C}\_{kj}^{nl}\left\{\boldsymbol{u}\_{j}(\mathbf{x},t)\boldsymbol{n}\_{m}(\mathbf{x})\delta\_{\mathbb{S}}(\mathbf{x})H(t)\right\}, \end{aligned}$$

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

$$
\hat{w}\_i(\mathbf{x}, t) = \mathbf{U}\_i^k \ast \hat{\mathbf{G}}\_k + \hat{\mathbf{U}}\_i^k \underset{\mathbf{x}}{\ast} u\_k^1(\mathbf{x}) H\_S^-(\mathbf{x}) + \partial\_t \hat{\mathbf{U}}\_i^k \underset{\mathbf{x}}{\ast} u\_k^0(\mathbf{x}) H\_S^-(\mathbf{x}) + 
$$

$$
+ \hat{\mathbf{U}}\_i^k \ast \mathbf{g}\_k(\mathbf{x}, t) \delta\_S(\mathbf{x}) H(t) - \mathbf{C}\_{k\bar{\mathbf{j}}}^{m\bar{l}} \hat{\mathbf{U}}\_i^k \ast \left( u\_j(\mathbf{x}, t) n\_m(\mathbf{x}) \delta\_S(\mathbf{x}) H(t) \right)\_l. \tag{45}
$$

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

$$\mathbf{C}\_{kj}^{ml} \, \partial\_t \hat{V}\_i^k \ast \left( u\_j n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) H(t) \right), \\ = \mathbf{C}\_{kj}^{ml} \, \partial\_t \hat{V}\_i^k \ast \left( u\_j n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) H(t) \right), \\ = \mathbf{C}\_{kj}^{ml} \, \partial\_t \hat{V}\_i^k \ast \left( u\_j n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) H(t) + u\_j^0(\mathbf{x}) n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) \delta(t) \right) \\ = \mathbf{C}\_{kj}^{ml} \, \partial\_t \hat{V}\_i^k \ast u\_j n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) H(t) + \mathbf{C}\_{kj}^{ml} \, \partial\_t \hat{V}\_i^k \ast \limits\_{\mathbf{x}} u\_j^0(\mathbf{x}) n\_m(\mathbf{x}) \delta\_\mathbf{S}(\mathbf{x}) $$

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>

$$(\hat{w}\_i, \rho\_i) = \left(\hat{U}\_i^k \ast \hat{F}\_k, \rho\_i\right) = \left(\hat{U}\_i^k \ast L\_{kj}(\partial\_\mathbf{x}, \partial\_t)\hat{u}\_j, \rho\_i\right) = $$

$$= \left(L\_{kj}(\partial\_\mathbf{x}, \partial\_t)\hat{U}\_i^k \ast \hat{u}\_j, \rho\_i\right) = \left(\delta\_i^j \delta(\mathbf{x}, \mathbf{t}) \ast \hat{u}\_j, \rho\_i\right) = (\hat{u}\_i, \rho\_i).$$

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 follows that *w*^*i*ð Þ¼ *x*, *t* 0, *x* �∈ *D*�. This implies the assertion of the theorem, since the solution of the problem is unique.

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.

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

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.
