**7. Singular boundary integral equations of plane boundary value problems**

Let us consider the solutions of the posed boundary value problems in the case *N* = 2. For the integral representation of the dynamic analogue of the Green formula, we also calculate for the Green function (29):

$$
\hat{W} = \frac{1}{2\pi} d\_0(r, t) \, \ast \, H(t) = \frac{1}{2\pi c} d\_1(r, t), \tag{35}
$$

*for in form of*

*DOI: http://dx.doi.org/10.5772/intechopen.91693*

*ε*-vicinity at point *x*\*. We denote

the area bounded by a compound path :

differential in polar coordinates: , therefore

We have the last equality due to the inequality:

**49**

**Proof.** In the flat case, the formula of Theorem 3, taking into account the

*Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon…*

Substituting the form of the cores (29), (6.2), we obtain the first formula of the

In this equality, we pass to the limit for . For , arc length

. Because the outside

theorem. For , all integrals are convergent, because , and *U* has an integrable singularity at the front (30). Let us prove the second formula for *.* We write a dynamic analogue of Green's formula for a region with a puncture

carriers of the cores, can be written in the integral form**:**

$$
\hat{H}(\mathbf{x}, t, n) = \frac{1}{2\pi c} d\_2(r, t) \frac{\hat{\mathcal{O}r}}{\hat{\mathcal{O}n}}, \quad r = \|\mathbf{x}\|, \tag{36}
$$

where

$$\begin{aligned} \dot{d}\_{0}(r,t) &= H\left(ct - r\right) \frac{c\dot{h}\left(m\sqrt{c^{2}t^{2} - r^{2}}\right)}{\sqrt{c^{2}t^{2} - r^{2}}},\\ \dot{d}\_{1}(r,t) &= H\left(ct - r\right) \int\_{0}^{\sqrt{c^{2}t^{2} - r^{2}}} \frac{c\dot{h}\left(mz\right)}{\sqrt{z^{2} + r^{2}}} dz,\\ \dot{d}\_{2}(r,t) &= -H(ct - r) \left[\frac{1}{r} - \frac{c\dot{h}\left(m\sqrt{c^{2}t^{2} - r^{2}}\right)}{ct}\right] - H(ct - r)r \int\_{0}^{\sqrt{c^{2}t^{2} - r^{2}}} \frac{c\dot{h}\left(m\overline{z}\right)}{\left(\sqrt{z^{2} + r^{2}}\right)^{3}} dz. \end{aligned}$$

Consider the values of these functions at the front *r = ct, t > 0*. From (29), (27) follows that

$$\left.W\right|\_{r=ct} = 0, \quad \left.H\right|\_{r=ct} = 0,\tag{37}$$

Consequently, unlike from *U*, *W* and *H* are continuous at the front. When , we have an asymptotic representation:

$$U = \frac{ch(mct)}{2\pi ct} + O(r), \quad H = -\frac{1}{2\pi cr} \frac{\partial r}{\partial n} + O(1) \tag{38}$$

Now we turn to the integral notation of the dynamic analogue of Green's formula for *N* = 2.

**Theorem 6.** *The solution of the initial-boundary value problem for the KG-equation in the flat case is representable: for in form of*

$$2\pi \, u\left(x, t\right)H(t) = \int\_{S\_t(x)} H\left(ct - r\right)dS(y)\int\_{r/c}^t \left\{d\_2\left(r, \tau\right)\frac{\partial r}{\partial n(y)}\dot{u}\left(y, t - \tau\right)\right\}d\tau - \int\_{S\_t(x)} H\left(ct - r\right)\dot{S}\_t\left(r\right)\frac{\partial \left(r\right)}{\partial n(y)}\dot{u}\left(y, t - \tau\right)d\tau + \int\_{S\_t(x)}^t \left\{d\_2\left(r, \tau\right)\frac{\partial \left(r\right)}{\partial n(y)}\dot{u}\left(y, t - \tau\right)\right\}d\tau$$

$$-\int\_{S\_t(x)} H\left(ct - r\right)dS(y)\int\_{r/c}^t \dot{d}\_0\left(r, \tau\right)p\left(y, t - \tau\right)d\tau, \quad r = \left\|y - x\right\|;$$

*Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon… DOI: http://dx.doi.org/10.5772/intechopen.91693*

*for in form of*

construct the solution, one should use the formula in ultraprecise form (23). We

Let us consider the solutions of the posed boundary value problems in the case

Consider the values of these functions at the front *r = ct, t > 0*. From (29), (27)

Consequently, unlike from *U*, *W* and *H* are continuous at the front.

Now we turn to the integral notation of the dynamic analogue of Green's

**Theorem 6.** *The solution of the initial-boundary value problem for the KG-equation*

When , we have an asymptotic representation:

*in the flat case is representable: for in form of*

ð35Þ

ð36Þ

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**7. Singular boundary integral equations of plane boundary value**

*N* = 2. For the integral representation of the dynamic analogue of the Green

construct solutions to initial-boundary value problems.

*Mathematical Theorems - Boundary Value Problems and Approximations*

formula, we also calculate for the Green function (29):

**problems**

where

follows that

formula for *N* = 2.

**48**

$$
\pi c u(\infty, t) H(t) = V.P. \int\_{S\_r(\varepsilon)} H\left(ct - r\right) \frac{\partial r}{\partial n(\mathcal{Y})} \, dS(\mathcal{Y}) \int\_{\tau/c}^t d\_2\left(r, \tau\right) \dot{u}\left(\mathcal{Y}, t - \tau\right) d\tau - \int\_{S\_r(\varepsilon)}^t H\left(ct - r\right) \dot{u}\left(\mathcal{Y}, t - \tau\right) d\tau.
$$

$$
$$

**Proof.** In the flat case, the formula of Theorem 3, taking into account the carriers of the cores, can be written in the integral form**:**

$$2\pi\hat{u}\left(x,t\right) = \int\_{S\_t\left(x\right)} H\left(ct-r\right)dS(y)\int\_{r\wedge c}^t H\left(x,y,\tau,n(y)\right)\hat{u}\left(y,t-\tau\right)d\tau - \int\_{S\_t\left(x\right)} H\left(ct-r\right)\left(y,t-\tau\right)d\tau - \int\_{\mathbb{R}^{d\times d}} H\left(ct-r\right)\hat{u}\left(y,t-\tau\right)d\tau.$$

$$-H\left(t\right)\int\_{S\_t\left(x\right)} H\left(ct-r\right)dS(y)\int\_{r\wedge c}^t U\left(x,y,\tau\right)\frac{\hat{\sigma}u\left(y,t-\tau\right)}{\hat{\sigma}n(y)}d\tau.$$

Substituting the form of the cores (29), (6.2), we obtain the first formula of the theorem. For , all integrals are convergent, because , and *U* has an integrable singularity at the front (30). Let us prove the second formula for *.*

We write a dynamic analogue of Green's formula for a region with a puncture *ε*-vicinity at point *x*\*. We denote . Because the outside

the area bounded by a compound path :

$$\begin{aligned} \boldsymbol{\Sigma}\_{1\varepsilon} + \boldsymbol{\Sigma}\_{2\varepsilon} &= -\int\_{\Omega\_{\varepsilon}(\boldsymbol{x}^\*)} \boldsymbol{H} \left( c\boldsymbol{t} - \boldsymbol{r} \right) \frac{\partial \boldsymbol{r}}{\partial n(\boldsymbol{y})} dS(\boldsymbol{y}) \int\_{\boldsymbol{r}\neq\boldsymbol{t}}^{\boldsymbol{t}} d\_2 \left( r, \boldsymbol{\tau} \right) \dot{\boldsymbol{n}} \left( \boldsymbol{y}, \boldsymbol{t} - \boldsymbol{\tau} \right) \frac{d\boldsymbol{\tau}}{\boldsymbol{c}} - \\ &- \int\_{\Omega\_{\varepsilon}(\boldsymbol{x}^\*)} \boldsymbol{H} \left( c\boldsymbol{t} - \boldsymbol{r} \right) dS(\boldsymbol{y}) \int\_{\boldsymbol{r}/\boldsymbol{c}}^{\boldsymbol{t}} d\_0 \left( r, \boldsymbol{\tau} \right) p \left( \boldsymbol{y}, \boldsymbol{t} - \boldsymbol{\tau} \right) d\boldsymbol{\tau} = 0 \end{aligned}$$

In this equality, we pass to the limit for . For , arc length differential in polar coordinates: , therefore

$$\lim\_{\varepsilon \to 0} \int\_{\Gamma\_k^{\varepsilon}(\varepsilon^{\theta})} H(c\tau - \varepsilon) d\mathbf{S}(\mathbf{y}) \int\_{\varepsilon/c}^{\varepsilon} d\_0(\varepsilon, \tau) p\left(\mathbf{y}, t - \tau\right) d\tau = $$
 
$$= \lim\_{\varepsilon \to 0} \varepsilon \pi \int\_{\varepsilon/c}^{\varepsilon} \left\{ \frac{ch(mc\tau)}{c\tau} p\left(\mathbf{x}^{\theta}, t - \tau\right) \right\} d\tau = \pi c \lim\_{\varepsilon \to 0} \varepsilon \int\_{\varepsilon}^{\varepsilon/c} \left\{ \frac{ch(m\varepsilon\tau)}{c\tau} p\left(\mathbf{x}^{\theta}, t - \frac{\varepsilon}{c}\tau\right) \right\} d\tau = 0$$

We have the last equality due to the inequality:

$$\left| \int\_{1}^{t/\varepsilon} \left\{ \frac{ch(m\varepsilon\tau)}{\tau} p\left(\chi^{\ast}, t - \frac{\varepsilon}{c}\tau\right) \right\} d\tau \right| \leq ch(mct) \int\_{0}^{\infty} \left| p\left(\chi^{\ast}, \tau\right) \right| d\tau$$

Consider the limit of the first integral:

$$\lim\_{\varepsilon \to 0} \Sigma\_{|\varepsilon} = \lim\_{\varepsilon \to 0} \int\_{\Omega\_{\varepsilon}(r^\*)} H(ct - r) \frac{\partial r}{\partial n(\mathbf{y})} dS(\mathbf{y}) \int\_{r^\* | \varepsilon}^{\dagger} d\_2(r, \tau) \dot{\mu}(\mathbf{y}, t - \tau) \frac{d\tau}{c} =$$

$$= V.P.\int\_{S\_{\varepsilon}(r^\*)} H(ct - r) \frac{\partial r}{\partial n(\mathbf{y})} dS(\mathbf{y}) \int\_{r^\* | \varepsilon}^{\dagger} d\_2(r, \tau) \dot{\mu}(\mathbf{y}, t - \tau) \frac{d\tau}{c} \tag{39}$$

$$= \lim\_{\varepsilon \to 0} \int\_{\Gamma\_{\varepsilon}(r^\*)} H(ct - r) \frac{\partial r}{\partial n(\mathbf{y})} dS(\mathbf{y}) \int\_{r^\* | \varepsilon}^{\dagger} d\_2(r, \tau) \dot{\mu}(\mathbf{y}, t - \tau) \frac{d\tau}{c} = I\_S + \lim\_{\varepsilon \to 0} I\_{\varepsilon}$$

where

*the form:*

**51**

The values of these functions at the front :

*DOI: http://dx.doi.org/10.5772/intechopen.91693*

Let us impose the notation for the shift functions:

following asymptotics are true:

Consequently, *W* is continuous at the wave front; the last term in the representation *H* has a discontinuity of the first kind at the front. When , the

*Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon…*

**Theorem 7**. *The generalized solution of boundary value problems for a homogeneous KG-equation satisfying zero initial conditions is representable in*

*For , the last integral is singular, taken in the sense of the principal value.* **Proof.** Using the conditions of Theorem and (30), we write in this case a dynamic analogue of Green's formula (22). We compute convolution sequentially:

ð41Þ

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We calculate the last limit on the right side.

Since on :

$$H = \frac{1}{2\pi c\varepsilon} + O(1),$$

Therefore

$$\lim\_{\varepsilon \to 0} I\_{\varepsilon} = -\lim\_{\varepsilon \to 0} \int\_{\Gamma\_{\varepsilon}^{-}(\chi^{\mathbf{a}})} \frac{\widehat{\partial}r}{\widehat{\partial}n(\chi)} dS(\chi) \int\_{\varepsilon/c}^{\prime} d\_{2}\left(\mathfrak{x},\tau\right) \dot{n}\left(\chi^{\mathbf{a}},t-\tau\right) \frac{d\tau}{c} = 0$$

$$= \lim\_{\varepsilon \to 0} \pi \varepsilon \int\_{\varepsilon/c}^{\prime} \frac{\dot{n}\left(\chi^{\mathbf{a}},t-\tau\right)}{\varepsilon} d\tau = \pi \lim\_{\varepsilon \to 0} \int\_{0}^{\prime} \dot{n}\left(\chi^{\mathbf{a}},t-\tau\right) d\tau = $$

$$= \pi \left(\mu\left(\chi^{\mathbf{a}},0\right) - \mu\left(\chi^{\mathbf{a}},t\right)\right) = -\pi \mu\left(\chi^{\mathbf{a}},t\right)$$

Adding, taking into account the last equality, we obtain the second formula of the theorem. The theorem has been proved.

In the case of the first boundary value problem, the left side of the equation of Theorem 6.1 and the first integral on the right are known, determined by the boundary conditions. Solving it, we determine the normal derivative of the desired function on the boundary, after which the formula of the theorem allows us to calculate the solution at any point in the domain of definition. In the case of the second boundary-value problem, we have a BIE to determine the unknown boundary values of the unknown function *u* from the boundary values of its normal derivative. Solving it, we determine its values at the border, after which we determine the solution.
