**4. The dynamic analogue of Green's formula with constant scattering potential**

Consider the case when scattering potential is constant:

$$q(\mathbf{x}) = \pm m^2$$

To build the solution of BVP, we move to the space of generalized functions. To do this, we introduce the characteristic function of the solution domain

$$H\_D^-(\mathfrak{x}, t) \equiv H\_S^-(\mathfrak{x}) H(t),$$

where *H*� *<sup>S</sup>* ð Þ *x* is a characteristic function of set *S*�, which is equal to 0.5 on its boundary *S*; and *H t*ð Þ is Heaviside's function, which is equal to 0.5 at *t* = 0. *H*� *<sup>D</sup>* is a characteristic function of space-time cylinder *D*�. It is easy to show that:

$$\frac{\partial H\_D^-}{\partial \mathbf{x}\_j} = -n\_j \delta\_\mathcal{S}(\mathbf{x}) H(t), \quad \frac{\partial H\_D^-}{\partial t} = -n\_j H\_\mathcal{S}^-(\mathbf{x}) \delta(t), \tag{17}$$

where *δ*ð Þ*t* is singular Dirac's function.

To use the methods of the theory of generalized functions, we define the solution by zero outside the domain of the solution of the boundary value problem. For this, we put regular generalized functions:

$$
\hat{\mu} = \mathfrak{u}(\mathfrak{x}, t) H\_D^-(\mathfrak{x}, t), \quad \hat{f} = f(\mathfrak{x}, t) H\_D^-(\mathfrak{x}, t), \tag{18}
$$

where *u x*ð Þ , *t* is the classical solution of the BVP.

Consider the action of the KG-operator on *u*^. Since ½ � *u <sup>S</sup>* ¼ �*u*, and performing generalized differentiation using (17), we get

$$
\Box\_{\boldsymbol{c}} \hat{\boldsymbol{u}} \pm m^2 \hat{\boldsymbol{u}} = -\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{n}} \delta\_{\mathcal{S}}(\boldsymbol{x}) \boldsymbol{H}(t) - \boldsymbol{H}(t) \left( \boldsymbol{u} \boldsymbol{n}\_j \delta\_{\mathcal{S}}(\boldsymbol{x}) \right)\_j - \boldsymbol{c}^{-2} \boldsymbol{H}\_{\mathcal{S}}^{-}(\boldsymbol{x}) \boldsymbol{u}\_0(\boldsymbol{x}) \dot{\delta}(t)
$$

$$
\tag{19}
$$

where *δS*ð Þ *x H t*ð Þ is simple layer on lateral surface of a space-time cylinder *D* ¼ *S* � *R*<sup>þ</sup> f g.

Note that the densities of simple and double layers here are determined by the boundary conditions, some of which (depending on the boundary value problem) are known, and the given initial conditions.

*u*\_ *<sup>S</sup>*ð Þ¼ *x*, 0 *v*0ð Þ *x* , *x*∈*S*, (15)

� (16)

which is typical for physical tasks. In this case, at the initial moment of time, a shock front is formed at the boundary *S*, which propagates with a velocity *с* in *S*�. To construct continuously differentiable solutions, this condition is necessary. Here we will not enter it. Here, we not enter it and suppose that *u*0ð Þ *x* ∈*C S*� ð Þ þ *S* , *v*0ð Þ *x* ∈ *L*<sup>1</sup> *S*� ð Þ þ *S* , *p x*ð Þ , *t* ∈ *L*1ð Þ *D* , and *uS*ð Þ *x*, *t* a Holder's function on *S*: ∀*β*, 0<*β* ≤1,

*Mathematical Theorems - Boundary Value Problems and Approximations*

*uS*ð Þ� *<sup>x</sup>*, *<sup>t</sup> uS*<sup>ð</sup> *<sup>y</sup>*, *<sup>t</sup>*Þj≤*const x*k k � *<sup>y</sup> <sup>β</sup>* , �

here *L*1ð Þ … is the Lebeg's space of summable on the specified set of functions. Let us mark as *D* ¼ *S* � *R*<sup>þ</sup> f g, the lateral surface of the space-time cylinder is

**Theorem 1. (***Energy conservation law***).** *If u*(*x,t*) *is classic solution of edge prob-*

2 ð

**Proof.** We integrate the energy Eq. (9) over a field with allowance for the partition of the field of integration by *Fk* wave fronts. Note that the first two terms can be considered as the divergence of the corresponding vector in space *R<sup>N</sup>*þ<sup>1</sup>

, we get

, *tdV x*ð Þ� *; t*

2

*c* �2

Hereinafter, we denote *dV x*ð Þ¼ *dx*<sup>1</sup> … *dxN*, *dV x*ð Þ¼ , *t dV x*ð Þ*dt*; *dFk*ð Þ *x*, *t* is the differential of the surface area at the corresponding point of the wave front. By

¼ �k k*ν Nc*

Therefore the last integral is zero. Taking into account the notation for the boundary functions, we get the formula of the theorem. From this theorem follows

ð

*Fk*

<sup>0</sup>ð Þ *<sup>x</sup>* � � � �

which is continuous in the regions between the fronts. Therefore, using the

ð

*D*�

*Fk*

*Fk*

<sup>ð</sup>*E x*ð Þ� *; <sup>t</sup> <sup>E</sup>*0ð Þ *<sup>x</sup>* Þ þ <sup>1</sup>

*dS x*ð Þ*dt* <sup>þ</sup><sup>X</sup>

*Eν<sup>t</sup>* � *uu*\_ , *<sup>j</sup>ν<sup>j</sup>* � � *S*�

2 ð*t*

0 *dt* ð

ð

*u u*\_ , *<sup>j</sup>* � �, *jdV x*ð Þ *; <sup>t</sup>*

ð Þ� *<sup>x</sup>; <sup>t</sup> <sup>u</sup>*<sup>2</sup>

*Fk*

*∂u ∂n*

� �

*dV x*ð Þ�

*dFk*ð Þ¼ *x; t* 0

¼ 0*:*

*D*�

*<sup>E</sup>ν<sup>t</sup>* � *<sup>∂</sup><sup>u</sup> ∂ν u*\_

�<sup>1</sup> *<sup>E</sup>* <sup>þ</sup> *cu*\_

� �

*uf x* \_ ð Þ *; t dV x*ð Þþ *; t*

*q x*ð Þ *<sup>u</sup>*<sup>2</sup>

*S*�

*q x*ð Þ *<sup>u</sup>*<sup>2</sup>

ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>2</sup> <sup>0</sup>ð Þ *<sup>x</sup>* � � *dV x*ð Þ

*f x*ð Þ , *t u x* \_ð Þ , *t dV x*ð Þ

,

such that for ∀*x*∈*S*, *y*∈ *S*, *t*≥0

*D*� ¼ *S*� � *R*þ, *R*<sup>þ</sup> ¼ ð Þ 0, þ∞ .

ð Þ *E x*ð Þ� , *t E*0ð Þ *x dV x*ð Þþ 0*:*5*c*

ð Þ *u*\_ *<sup>S</sup>*ð Þ *x*, *t p*ð*x*, *t*Þ *dS x*ð Þ� *c*

ð

*S*�

¼ *c* 2 ð*t*

0 *dt* ð

Ostrogradsky-Gauss theorem in *R<sup>N</sup>*þ<sup>1</sup>

*c* �2 *E* þ 1 2 *q x*ð Þ*u*<sup>2</sup>

*c* �2

*u*\_ *∂u ∂n* � �

> *c* �2

� �

*uf x* \_ ð Þ *; t dV x*ð Þ¼ *; t*

ð

*D*�

*D*�

*S*�

ð

*S*

þ ð

þ ð

� ð*t*

0

virtue of (3) and Lemma 2,

the Theorem 2.

**42**

*S*

*lem, then*

The solution of Eq. (19) is convolution of the right part of the equation with its fundamental solution *U x* ^ ð Þ , *<sup>t</sup>* , satisfying the conditions:

$$
\Box\_c \hat{U} \pm m^2 \hat{U} = \delta(\mathfrak{x})\delta(t),
\tag{20}
$$

*<sup>u</sup>*^ ¼ � *<sup>U</sup>*^ <sup>∗</sup> *<sup>∂</sup><sup>u</sup>*

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

� *c*

� *<sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup>

*∂n*

�<sup>2</sup> *U*^ ∗

*<sup>U</sup>*^ <sup>∗</sup> *unjδS*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>j</sup>* <sup>¼</sup> *<sup>W</sup>*^ , *<sup>t</sup>* <sup>∗</sup> *unjδS*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>j</sup>*

regularization of under integral functions on fronts.

2.BVP at zero initial conditions and *f x*ð Þ¼ , *t* 0.

□*cu*^ � *<sup>m</sup>*<sup>2</sup>

solutions of two BV problems:

**for** *N* **= 2**

**45**

1.Cauchy problem at *f x*ð Þ , *t* 6¼ 0;

boundary problem in the notation used here.

Since

*<sup>x</sup> <sup>H</sup>*�

the derivative of a generalized function and the continuity of *u*, that

*<sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup> *u x*ð Þ *;* <sup>0</sup> *nj*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> <sup>δ</sup>*ðÞ¼ *<sup>t</sup> <sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup>

putting these ratios in (22), we obtain the formula of the theorem.

tion, these formulas may be called *dynamical analog of Green's formula.*

From Theorem 3, it is consequent that the solution of the problem is entirely defined by initial data, boundary means of normal derivative of function *u x*ð Þ , *t* , and its speed *<sup>u</sup>*\_ <sup>¼</sup> *<sup>u</sup>*, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>tu*. By analog with representation of Laplace's equation solu-

Formula (27) of Theorem 3 allows at once to go to its integral writing without

By virtue of linearity of equations, solutions of BVPs may be obtained as a sum

**5. The generalized solution of the Cauchy problem for the KG-equation**

Let us consider the Cauchy problem for the KG equation of the below form:

*<sup>u</sup>*^ <sup>¼</sup> ^*f x*ð Þ , *<sup>t</sup>* , *<sup>x</sup>*<sup>∈</sup> *RN*, *<sup>t</sup>* <sup>&</sup>gt;0, (28)

of solutions of these two problems with correction of boundary conditions for second problem with account of boundary meanings of Cauchy problem solutions.

Solution of Cauchy problem for that equation has been early obtained by Vladimirov (see [9]). We get it here for the complete solution of the initial-

Then let us consider representation of solution of edge problem for Klein-Gordon equations in spaces with dimensions *N* = 2,3, characterized for mathematical physics problems. To avoid complexity of formulas under building of integral representation of dynamical analog of Green's formula, let us consider consequently

*<sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ� *<sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup> *un*\_ *<sup>j</sup>*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ð Þ�

�2 *U*^ ∗ *<sup>x</sup> <sup>H</sup>*�

<sup>¼</sup> *<sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup> *u x* \_ð Þ *; <sup>t</sup> nj*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞþ *<sup>W</sup>*^ , *<sup>j</sup>* <sup>∗</sup> *u x*ð Þ *;* <sup>0</sup> *nj*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> <sup>δ</sup>*ð Þ*<sup>t</sup>*

*<sup>x</sup> <sup>u</sup>*0ð Þ *<sup>x</sup> nj*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup>* ,

*<sup>S</sup>* ð Þ *x u*\_ <sup>0</sup>ð Þ�*x*

(27)

*<sup>x</sup> <sup>u</sup>*0ð Þ *<sup>x</sup> nj*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ� *<sup>x</sup> <sup>c</sup>*

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

*<sup>S</sup>* ð Þ *x u*0ð Þ *x* , *<sup>t</sup>* <sup>þ</sup> ^*<sup>f</sup>* <sup>∗</sup> *<sup>U</sup>*^

**Proof.** Let us consider the formula (22). It is easy to show, using the definition of

*un <sup>j</sup>δS*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>t</sup>* <sup>¼</sup> *u x* \_ð Þ , *<sup>t</sup> <sup>n</sup> <sup>j</sup>*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞþ *u x*ð Þ , 0 *<sup>n</sup> <sup>j</sup>*ð Þ *<sup>x</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> <sup>δ</sup>*ð Þ*<sup>t</sup>*

Using this equality and the convolution differentiation property [1], we have

and *radiation conditions*:

$$
\hat{U}(\mathbf{x},t) = \mathbf{0} \text{ at } t < \mathbf{0},
\hat{U}(\mathbf{x},t) = \mathbf{0} \text{ at } ||\mathbf{x}|| > ct. \tag{21}
$$

Let us call it the *Green's function* of Eq. (1).

The solution of (19) will be obtained in the form of the following convolution of right part of (19) and Green's function, which is equal to

$$\begin{split} \hat{u} \triangleq \boldsymbol{u}(\mathbf{x},t)H\_{\mathcal{S}}^{-}(\mathbf{x})H(t) &= -\hat{U} \ast \frac{\partial \boldsymbol{u}}{\partial t} \delta\_{\mathcal{S}}(\mathbf{x})H(t) - \left(\hat{U} \ast \boldsymbol{u}\boldsymbol{u}\_{\hat{f}} \delta\_{\mathcal{S}}(\mathbf{x})H(t)\right)\_{\cdot,\cdot} - c^{-2} \Big(\hat{U} \ast H\_{\mathcal{S}}^{-}(\mathbf{x})\boldsymbol{u}\_{0}(\mathbf{x})\Big)\_{\cdot,\cdot} \\ &- c^{-2} \hat{U} \ast H\_{\mathcal{S}}^{-}(\mathbf{x})\dot{\boldsymbol{u}}\_{0}(\mathbf{x}) + \hat{f}(\mathbf{x},t) \ast \hat{U} \end{split} \tag{22}$$

Here the symbol "\*" means that convolution is taken only by *x*. Moreover, the solution is unique in the class of functions that allows convolution with *U*. Hence, it is easy to obtain a solution to the Cauchy problem (in the absence of *<sup>S</sup>*, *<sup>S</sup>*� <sup>¼</sup> *<sup>R</sup><sup>N</sup>*).

**Consequence 1.** The generalized solution of the Cauchy problem has the form:

$$\hat{u}(\mathbf{x},t) = -c^{-2}\hat{U} \underset{\mathbf{x}}{\*} \dot{u}\_0 - c^{-2} \left(\hat{U} \underset{\mathbf{x}}{\*} u\_0\right), \\ +\hat{f} \* \hat{U} \tag{23}$$

**Consequence 2.** At zero initial data and *f* ¼ 0, the generalized solution has the form:

$$\hat{u} = -\hat{U} \ast \frac{\partial u}{\partial n} \delta\_{\mathbb{S}}(\mathfrak{x}) H(t) - \hat{U} \,, \mathfrak{z} \ast \mathfrak{w}\_{\mathbb{S}} \delta\_{\mathbb{S}}(\mathfrak{x}) H(t) \tag{24}$$

Formulas (22) and (24) express the solution of boundary value problems through the boundary values of the unknown function and its derivative along the normal to the boundary, i.e., they are similar to the Green formula for solutions of elliptic equations [9]. However, due to the singularities of the fundamental solutions of hyperbolic equations on the wave front, the form of which depends on the dimension of space, their integral representation gives divergent integrals containing derivatives of the fundamental solution. To construct regular integral representations, we introduce an antiderivative function:

$$
\hat{\mathcal{W}} = \hat{U} \* \delta(\mathfrak{x}) H(\mathfrak{t}) = \hat{U} \underset{\mathfrak{t}}{\*} H(\mathfrak{t}) \Rightarrow \partial\_t \hat{\mathcal{W}} = \hat{U} \tag{25}
$$

and

$$
\hat{H}(\mathbf{x}, n, t) = \frac{\partial \hat{W}}{\partial \mathbf{x}\_j} n\_j = \frac{\partial \hat{W}}{\partial n} \tag{26}
$$

It is easy to see that *<sup>W</sup>*^ <sup>и</sup>*H*^ are also solutions (1) at ^*f x*ð Þ¼ , *<sup>t</sup> H t*ð Þ*δ*ð Þ *<sup>x</sup>* and ^*f x*ð Þ¼ , *<sup>t</sup> H t*ð Þ *<sup>∂</sup>δ*ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>∂</sup><sup>n</sup>* , respectively. The following theorem is true.

**Theorem 3.** *The generalized solution of boundary value problems has the form:* (*dynamic analogue of Green's formula*)

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

$$\begin{split} \hat{u} &= -\hat{U} \ast \frac{\partial u}{\partial n} \delta\_{\mathcal{S}}(\mathbf{x}) H(t) - \hat{W}\_{,j} \ast i \boldsymbol{n}\_{\hat{f}}(\mathbf{x}) \delta\_{\mathcal{S}}(\mathbf{x}) H(t) - \\ &- \hat{W}\_{,j} \underset{\stackrel{\text{\tiny}}{\text{\tiny x}}}{\text{\tiny x}} u\_{0}(\mathbf{x}) \boldsymbol{n}\_{\hat{f}}(\mathbf{x}) \delta\_{\mathcal{S}}(\mathbf{x}) - c^{-2} \hat{U} \underset{\stackrel{\text{\tiny}}{\text{\tiny x}}}{\text{\tiny x}} H\_{\mathcal{S}}^{-}(\mathbf{x}) \dot{u}\_{0}(\mathbf{x}) - \\ &- c^{-2} \Big( \hat{U} \underset{\stackrel{\text{\tiny}}{\text{\tiny x}}}{\text{\tiny x}} H\_{\mathcal{S}}^{-}(\mathbf{x}) u\_{0}(\mathbf{x}) \Big) \,, \_{t} + \hat{f} \ast \hat{U} \end{split} \tag{27}$$

**Proof.** Let us consider the formula (22). It is easy to show, using the definition of the derivative of a generalized function and the continuity of *u*, that

$$\left(\iota m\_j \delta\_\mathbb{S}(\mathbf{x}) H(t)\right)\_{\mathbf{t},t} = \dot{u}(\mathbf{x},t) n\_j(\mathbf{x}) \delta\_\mathbb{S}(\mathbf{x}) H(t) + u(\mathbf{x},\mathbf{0}) n\_j(\mathbf{x}) \delta\_\mathbb{S}(\mathbf{x}) \delta(t)$$

Using this equality and the convolution differentiation property [1], we have

$$\begin{aligned} \left( (\hat{\mathcal{U}} \* \boldsymbol{u}\mathfrak{n}\_{j} \delta\_{\mathbb{S}}(\mathbf{x}) H(\mathbf{t}) ) \right)\_{j} &= \left( \hat{\mathcal{W}}\_{\cdot, \mathfrak{t}} \* \boldsymbol{u}\mathfrak{n}\_{j} \delta\_{\mathbb{S}}(\mathbf{x}) H(\mathbf{t}) \right)\_{j} \\ &= \hat{\mathcal{W}}\_{\cdot, j} \* \dot{\mathfrak{u}}(\mathbf{x}, \mathfrak{t}) \mathfrak{n}\_{j}(\mathbf{x}) \delta\_{\mathbb{S}}(\mathbf{x}) H(\mathbf{t}) + \hat{\mathcal{W}}\_{\cdot, j} \* \boldsymbol{u}(\mathbf{x}, \mathbf{0}) \mathfrak{n}\_{j}(\mathbf{x}) \delta\_{\mathbb{S}}(\mathbf{x}) \delta(\mathbf{t}) \end{aligned}$$

Since

The solution of Eq. (19) is convolution of the right part of the equation with its

The solution of (19) will be obtained in the form of the following convolution of

*<sup>S</sup>* ð Þ *<sup>x</sup> <sup>u</sup>*\_ <sup>0</sup>ð Þþ *<sup>x</sup>* ^*f x*ð Þ *; <sup>t</sup>* <sup>∗</sup> *<sup>U</sup>*^

Here the symbol "\*" means that convolution is taken only by *x*. Moreover, the solution is unique in the class of functions that allows convolution with *U*. Hence, it is easy to obtain a solution to the Cauchy problem (in the absence of *<sup>S</sup>*, *<sup>S</sup>*� <sup>¼</sup> *<sup>R</sup><sup>N</sup>*). **Consequence 1.** The generalized solution of the Cauchy problem has the form:

*<sup>x</sup> <sup>u</sup>*\_ <sup>0</sup> � *<sup>c</sup>*

Formulas (22) and (24) express the solution of boundary value problems through the boundary values of the unknown function and its derivative along the normal to the boundary, i.e., they are similar to the Green formula for solutions of elliptic equations [9]. However, due to the singularities of the fundamental solutions of hyperbolic equations on the wave front, the form of which depends on the

dimension of space, their integral representation gives divergent integrals containing derivatives of the fundamental solution. To construct regular integral

**Consequence 2.** At zero initial data and *f* ¼ 0, the generalized solution has the

*<sup>∂</sup><sup>n</sup> <sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *<sup>U</sup>*^ <sup>∗</sup> *unjδS*ð Þ *<sup>x</sup> H t*ð Þ , *<sup>j</sup>* � *<sup>c</sup>*

�<sup>2</sup> *U*^ ∗

*<sup>x</sup> <sup>u</sup>*<sup>0</sup> 

*<sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *<sup>U</sup>*^ , *<sup>j</sup>* <sup>∗</sup> *unjδS*ð Þ *<sup>x</sup> H t*ð Þ (24)

*<sup>t</sup> H t*ðÞ) *<sup>∂</sup>tW*^ <sup>¼</sup> *<sup>U</sup>*^ (25)

*U x* ^ ð Þ¼ , *<sup>t</sup>* 0 at *<sup>t</sup>*<0, *U x* ^ ð Þ¼ , *<sup>t</sup>* 0 at k k*<sup>x</sup>* <sup>&</sup>gt;*ct:* (21)

*<sup>U</sup>*^ <sup>¼</sup> *<sup>δ</sup>*ð Þ *<sup>x</sup> <sup>δ</sup>*ð Þ*<sup>t</sup>* , (20)

�<sup>2</sup> *U*^ ∗

*<sup>x</sup> <sup>H</sup>*�

, *<sup>t</sup>* <sup>þ</sup> ^*<sup>f</sup>* <sup>∗</sup> *<sup>U</sup>*^ (23)

*<sup>S</sup>* ð Þ *x u*0ð Þ *x* 

(22)

(26)

, *t*

fundamental solution *U x* ^ ð Þ , *<sup>t</sup>* , satisfying the conditions:

*Mathematical Theorems - Boundary Value Problems and Approximations*

Let us call it the *Green's function* of Eq. (1).

*<sup>S</sup>* ð Þ *<sup>x</sup> H t*ðÞ¼� *<sup>U</sup>*^ <sup>∗</sup> *<sup>∂</sup><sup>u</sup>*

� *c* �2 *U*^ ∗ *<sup>x</sup> <sup>H</sup>*�

*u x* ^ð Þ¼� *; t c*

*<sup>u</sup>*^ ¼ �*U*^ <sup>∗</sup> *<sup>∂</sup><sup>u</sup>*

representations, we introduce an antiderivative function:

*<sup>W</sup>*^ <sup>¼</sup> *<sup>U</sup>*^ <sup>∗</sup> *<sup>δ</sup>*ð Þ *<sup>x</sup> H t*ðÞ¼ *<sup>U</sup>*^ <sup>∗</sup>

*H x* ^ ð Þ¼ , *<sup>n</sup>*, *<sup>t</sup>*

*∂W*^ *∂x <sup>j</sup>*

It is easy to see that *<sup>W</sup>*^ <sup>и</sup>*H*^ are also solutions (1) at ^*f x*ð Þ¼ , *<sup>t</sup> H t*ð Þ*δ*ð Þ *<sup>x</sup>* and

*<sup>∂</sup><sup>n</sup>* , respectively. The following theorem is true. **Theorem 3.** *The generalized solution of boundary value problems has the form:*

*<sup>n</sup> <sup>j</sup>* <sup>¼</sup> *<sup>∂</sup>W*^ *∂n*

�2 *U*^ ∗

*∂n*

right part of (19) and Green's function, which is equal to

and *radiation conditions*:

*u*^ ¼ *u x*ð Þ *; t H*�

form:

and

**44**

^*f x*ð Þ¼ , *<sup>t</sup> H t*ð Þ *<sup>∂</sup>δ*ð Þ *<sup>x</sup>*, *<sup>t</sup>*

(*dynamic analogue of Green's formula*)

□*cU*^ � *<sup>m</sup>*<sup>2</sup>

$$
\hat{\mathcal{W}}\_{\cdot j} \* \mathfrak{u}(\mathfrak{x}, \mathbf{0}) n\_{\mathfrak{j}}(\mathfrak{x}) \delta\_{\mathbb{S}}(\mathfrak{x}) \delta(t) = \hat{\mathcal{W}}\_{\cdot j} \*\_{\mathfrak{x}} \mathfrak{u}\_{0}(\mathfrak{x}) n\_{\mathfrak{j}}(\mathfrak{x}) \delta\_{\mathbb{S}}(\mathfrak{x}),
$$

putting these ratios in (22), we obtain the formula of the theorem.

From Theorem 3, it is consequent that the solution of the problem is entirely defined by initial data, boundary means of normal derivative of function *u x*ð Þ , *t* , and its speed *<sup>u</sup>*\_ <sup>¼</sup> *<sup>u</sup>*, *<sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>tu*. By analog with representation of Laplace's equation solution, these formulas may be called *dynamical analog of Green's formula.*

Formula (27) of Theorem 3 allows at once to go to its integral writing without regularization of under integral functions on fronts.

Then let us consider representation of solution of edge problem for Klein-Gordon equations in spaces with dimensions *N* = 2,3, characterized for mathematical physics problems. To avoid complexity of formulas under building of integral representation of dynamical analog of Green's formula, let us consider consequently solutions of two BV problems:

1.Cauchy problem at *f x*ð Þ , *t* 6¼ 0;

2.BVP at zero initial conditions and *f x*ð Þ¼ , *t* 0.

By virtue of linearity of equations, solutions of BVPs may be obtained as a sum of solutions of these two problems with correction of boundary conditions for second problem with account of boundary meanings of Cauchy problem solutions. Solution of Cauchy problem for that equation has been early obtained by Vladimirov (see [9]). We get it here for the complete solution of the initialboundary problem in the notation used here.
