**8. Dynamic analog of Green's formula for solutions of the KG-equation (***N* **= 3)**

To construct a dynamic analogue of Green's formula in integral form, we define и . By computing formulas (25)and (27), we obtain

$$4\pi c^2 \hat{W} = \frac{H\left(ct - r\right)}{r} - m\, f\_1(r, t), \qquad \qquad r = \left\|\infty\right\|\tag{40}$$

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

$$
\hat{H}\left(x, n, t\right) = -\frac{(x, n)}{2r} \left( \frac{\delta\left(ct - r\right)}{r} + \frac{H\left(ct - r\right)}{r^2} + 2m\hat{r}\_2\left(r, t\right) \right),
$$

$$
\text{where}
\quad f\_1(r, t) = H\left(ct - r\right) \int\_0^{\sqrt{c^2 r^2 - r^2}} \frac{J\_1\left(\frac{m}{c} z\right)}{\sqrt{z^2 + r^2}} dz,
$$

$$f\_2(r,t) = rH\left(ct - r\right) \int\_0^{\sqrt{c^2t^2 - r^2}} J\_1\left(\frac{m}{c}z\right) \, dz + \frac{rH\left(ct - r\right)}{ct\sqrt{c^2t^2 - r^2}} J\_1\left(\frac{m}{c}\sqrt{c^2t^2 - r^2}\right) \tag{41}$$

The values of these functions at the front :

$$f\_1(r, r/c) = 0, \quad f\_2\left(r, r/c\right) = \frac{m}{2c}$$

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 following asymptotics are true:

$$
\hat{W}\left(\mathbf{x},t\right) = \frac{H\left(t\right)}{8\pi c^2 r} + O(1) \tag{42}
$$

$$
\hat{H}\left(\mathbf{x},\mathbf{n},t\right) = \frac{\left(e\_{\mathbf{x}},n\right)}{8\pi c^{2}r^{2}}H\left(t\right) + O\left(r^{-1}\right) \tag{43}
$$

Let us impose the notation for the shift functions:

$$\begin{aligned} U\left(\mathbf{x},\mathbf{y},t\right) &= \hat{U}\left(\mathbf{x}-\mathbf{y},t\right), \; W\left(\mathbf{x},\mathbf{y},t\right) = \hat{W}\left(\mathbf{x}-\mathbf{y},t\right), \\ H\left(\mathbf{x},\mathbf{y},t,n\right) &= \hat{H}\left(\mathbf{x}-\mathbf{y},t,n\right) \end{aligned}$$

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

*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:

Consider the limit of the first integral:

*Mathematical Theorems - Boundary Value Problems and Approximations*

We calculate the last limit on the right side.

the theorem. The theorem has been proved.

Adding, taking into account the last equality, we obtain the second formula of

In the case of the first boundary value problem, the left side of the equation of

**8. Dynamic analog of Green's formula for solutions of the KG-equation**

и . By computing formulas (25)and (27), we obtain

To construct a dynamic analogue of Green's formula in integral form, we define

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 deter-

Since on :

Therefore

mine the solution.

**(***N* **= 3)**

**50**

ð39Þ

ð40Þ

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\begin{split} 4\pi \hat{U}(\mathbf{x},t) \* \frac{\partial u}{\partial n} \mathcal{S}\_{S}(\mathbf{x}) H(t) &= \int\_{S\_{r}(\mathbf{x})} \frac{1}{r} \frac{\partial u(\mathbf{y},t-r/c)}{\partial n(\mathbf{y})} dS(\mathbf{y}) - mc \Big[ \mathop{\rm div}\limits\_{S\_{\mathbf{r}}(\mathbf{x})} f\_{0}(r,\tau) \frac{\partial u(\mathbf{y},t-\tau)}{\partial n(\mathbf{y})} dS(\mathbf{y}) = \\ &\int\_{S\_{r}(\mathbf{x})} \frac{1}{r} \frac{\partial u(\mathbf{y},t-r/c)}{\partial n(\mathbf{y})} dS(\mathbf{y}) - m \int\_{S\_{\mathbf{r}}^{+}(\mathbf{x})} f\_{0}(r,r/c) \frac{\partial u(\mathbf{y},t-\tau)}{\partial n(\mathbf{y})} dV(\mathbf{y}); \\ &- 4\pi \hat{W}\_{\mathbf{r},f} \* \dot{u}(\mathbf{x},t) n\_{j}(\mathbf{x}) \delta\_{S}(\mathbf{x}) H(t) = - \int\_{S\_{\mathbf{r}}(\mathbf{x})} \dot{u}(\mathbf{y},t-r/c) \frac{(\mathbf{y} - \mathbf{x},\dot{n}(\mathbf{y}))}{c\tau^{2}} dS(\mathbf{y}) - \\ &- \int\_{S\_{\mathbf{r}}(\mathbf{x})} \frac{(\mathbf{y} - \mathbf{x},\dot{n}(\mathbf{y}))}{r^{3}} \dot{u}(\mathbf{y},t-\tau) \mathop{\rm dis}\limits\_{S\_{\mathbf{r}}(\mathbf{x})} \left[ \frac{(\mathbf{y} - \mathbf{x},\dot{n}(\mathbf{y}))}{c^{2}r} f\_{2}(r,\tau) \dot{u}(\mathbf{y},t-\tau) \mathbf{S}(\mathbf{y}). \right]; \end{split}$$

Moreover, the main value of the integral exists, because the integrand has a singularity of order on a two-dimensional surface *S*, function *u* is continuous on *S*, and the characteristic antisymmetric in opposite relative points. Let us consider the last limit. For , we have , ,

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

Passing in (44) to the limit in and transferring the last term to the right

Note that the constructed delayed singular BIE have a nonclassical type; since in addition to the boundary values *u*, *<sup>t</sup>* of the function and its normal derivative, the BIE includes a velocity that is unknown for the Dirichlet problem and is known for the Neumann problem. In addition, the integration region at the boundary depends on time, which also distinguishes these equations from the SEI for elliptic and parabolic problems. Solving the Dirichlet problem on the basis of the method of successive approximations, like elliptic problems, is impossible, since it requires the determination of boundary values of velocity. However, differentiation of generalized solutions on the boundary leads to hypersingular relations. This is a new class of BIE in delayed potentials, which requires a special study by the methods of functional analysis. However, to solve the resolving singular BIE that solve the boundary value problems, numerical boundary element method can be used.

This work was financially supported by the Ministry of Education and Science of

Formula ( ) gives a singular boundary integral equation for solving the second initial-boundary value problem. For the first boundary value problem, the unknown normal derivative falls under the sign of the surface integral with a

side, we obtain the formula of the theorem. The theorem has been proved.

weakly polar core. The remaining terms are known.

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

the Republic of Kazakhstan (grant AP05132272).

therefore

**9. Conclusion**

**Acknowledgements**

**53**

When , you can change the order of integration, therefore:

$$\int\_{0}^{t} d\tau \int\_{\mathbb{T}(\mathbf{x})} \frac{(\mathbf{y} - \mathbf{x}, n(\mathbf{y}))}{r^{3}} \dot{u}(\mathbf{y}, t - \tau) \, dS(\mathbf{y}) = -\int\_{\mathbb{S}(\mathbf{x})} u(\mathbf{y}, t - r/c) \frac{(\mathbf{y} - \mathbf{x}, n(\mathbf{y}))}{r^{3}} \, dS(\mathbf{y})$$

Summing up, we obtain the formula of the theorem.

Note that for points when , all cores have no singularities and the integrals on the right exist and define functions that are regular on a given set. Since regular functions are on left and right and they are equal on this set as generalized, by virtue of the du Bois-Reymond lemma [1], they are equal in the usual sense, like numerical functions.

Let us assume . We write a dynamic analogue of Green's formula for a region with a puncture vicinity point

Now let us move on to the limit . In the first integral, the integrand has a weak integrable singularity when *r* = 0. In the second integral, it does not have a singularity when *r* = 0. By virtue of this, the integrals over from these functions in the third and fourth term tend to zero. It is obvious that

$$\lim\_{\varepsilon \to 0} \int\_{S\_t(\chi) - O\_\delta(\chi)} u(\chi, t - r/c) \frac{(\chi - \chi, n(\chi))}{2r^3} dS(\chi) = V.P. \int\_{S\_t(\chi)} u\left(\chi, t - r/c\right) \frac{(\chi - \chi^\*, n(\chi))}{2r^3} dS(\chi),$$

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

Moreover, the main value of the integral exists, because the integrand has a singularity of order on a two-dimensional surface *S*, function *u* is continuous on *S*, and the characteristic antisymmetric in opposite relative points.

Let us consider the last limit. For , we have , , therefore

$$\lim\_{\varepsilon \to 0} \left| \left\{ \int\_{\Gamma\_{\varepsilon}^{-}(\mathbf{x}^{\*}, \boldsymbol{\mu})} u(\boldsymbol{\nu}, t - \boldsymbol{r} \, \boldsymbol{\nu}) \frac{(\boldsymbol{\nu} - \boldsymbol{x}^{\*}, \boldsymbol{\mu}(\boldsymbol{\nu}))}{r^{3}} dS(\boldsymbol{\nu}) \right\} \right| =$$

$$= \lim\_{\varepsilon \to 0} \left[ \int\_{\Gamma\_{\varepsilon}^{-}(\mathbf{x}^{\*}, t)} \left( u(\boldsymbol{\nu}, t - \boldsymbol{r} \, \boldsymbol{\nu} \, \boldsymbol{\nu}) - u(\boldsymbol{x}^{\*}, t) \right) \frac{(-\boldsymbol{\nu}(\boldsymbol{\nu}), \boldsymbol{\mu}(\boldsymbol{\nu}))}{\boldsymbol{\kappa}^{2}} dS(\boldsymbol{\nu}) \right) - \nu(\boldsymbol{x}^{\*}, t) \lim\_{\varepsilon \to 0} \left[ \int\_{\Gamma\_{\varepsilon}^{-}(\mathbf{x}^{\*}, t)} \frac{dS(\boldsymbol{\nu})}{\boldsymbol{\kappa}^{2}} \right] = -2\pi n \boldsymbol{\mu}(\boldsymbol{\nu}^{\*}, t) \int\_{\Gamma\_{\varepsilon}^{-}} \frac{\partial \mathcal{L}(\boldsymbol{\nu}^{\*}, t)}{\partial \boldsymbol{\kappa}^{2}} dS(\boldsymbol{\nu})$$

Passing in (44) to the limit in and transferring the last term to the right side, we obtain the formula of the theorem. The theorem has been proved.

Formula ( ) gives a singular boundary integral equation for solving the second initial-boundary value problem. For the first boundary value problem, the unknown normal derivative falls under the sign of the surface integral with a weakly polar core. The remaining terms are known.

## **9. Conclusion**

When , you can change the order of integration, therefore:

Note that for points when , all cores have no singularities and the integrals on the right exist and define functions that are regular on a given set. Since regular functions are on left and right and they are equal on this set as generalized, by virtue of the du Bois-Reymond lemma [1], they are equal in the usual sense, like

Let us assume . We write a dynamic analogue of Green's formula for a

Now let us move on to the limit . In the first integral, the integrand has a weak integrable singularity when *r* = 0. In the second integral, it does not have a singularity when *r* = 0. By virtue of this, the integrals over from these functions

in the third and fourth term tend to zero. It is obvious that

Summing up, we obtain the formula of the theorem.

*Mathematical Theorems - Boundary Value Problems and Approximations*

numerical functions.

**52**

region with a puncture vicinity point

Note that the constructed delayed singular BIE have a nonclassical type; since in addition to the boundary values *u*, *<sup>t</sup>* of the function and its normal derivative, the BIE includes a velocity that is unknown for the Dirichlet problem and is known for the Neumann problem. In addition, the integration region at the boundary depends on time, which also distinguishes these equations from the SEI for elliptic and parabolic problems. Solving the Dirichlet problem on the basis of the method of successive approximations, like elliptic problems, is impossible, since it requires the determination of boundary values of velocity. However, differentiation of generalized solutions on the boundary leads to hypersingular relations. This is a new class of BIE in delayed potentials, which requires a special study by the methods of functional analysis. However, to solve the resolving singular BIE that solve the boundary value problems, numerical boundary element method can be used.
