**3. Statement of non-stationary BVP for Klein-Gordon equation: Energy conservation law**

Let us construct the solution *u*(*x, t*) of Eq. (1) on a set *S*� ∈ *RN*, bounded by surface S, by *t*≥0*:* Lets introduce next marks: *n x*ð Þ is vector of external normal to *S*; *D* ¼ *S* � *R*<sup>þ</sup> f g is lateral surface of space-time cylinder *D*� ¼ *S*� � *R*þ, *<sup>R</sup>*<sup>þ</sup> <sup>¼</sup> ð Þ 0, <sup>þ</sup><sup>∞</sup> ; and the derivative of *<sup>u</sup>* on normal *<sup>n</sup>* at *<sup>i</sup>*, *<sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>n</sup>* ¼ *u*, *jn <sup>j</sup>*.

*Initial conditions:* At *t* ¼ 0 for *x*∈*S*�:

$$u(\mathbf{x}, \mathbf{0}) = u\_0(\mathbf{x}) \text{ for } \mathbf{x} \in \mathbb{S}^- + \mathcal{S} \tag{11}$$

$$\dot{u}(\mathbf{x}, \mathbf{0}) = v\_0(\mathbf{x}) \text{ for } \mathbf{x} \in \mathbb{S}^- \tag{12}$$

We consider two *boundary value problems* corresponding to the Dirichlet and Neumann conditions:

$$(\text{BVP I})\quad u(\mathbf{x},t) = u\_{\mathcal{S}}(\mathbf{x},t) \text{ for } \mathbf{x} \in \mathcal{S} \tag{13}$$

$$(\text{BVP II}) \quad \frac{\partial u}{\partial n} = p(\mathbf{x}, t) \text{ for } \mathbf{x} \in \mathbb{S} \tag{14}$$

At the shock wave fronts, the Hadamard conditions (5) and (6) on jumps are satisfied. Note that shock waves always occur if the condition of matching the initial and boundary data on the velocities is not satisfied

$$
\dot{u}\_{\mathcal{S}}(\mathbf{x}, \mathbf{0}) = \nu\_0(\mathbf{x}), \quad \mathbf{x} \in \mathcal{S}, \tag{15}
$$

**Theorem 2.** If *q x*ð Þ≥0*, then the classic solution of first (second) BVP for Klein-*

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

Since both terms are nonnegative, therefore, *E* = 0 and *u* = 0. The theorem is

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

*q x*ð Þ¼�*m*<sup>2</sup>

To build the solution of BVP, we move to the space of generalized functions. To

*<sup>S</sup>* ð Þ *x H t*ð Þ,

*<sup>D</sup>* is a

*<sup>S</sup>* ð Þ *x δ*ð Þ*t* , (17)

*<sup>D</sup>*ð Þ *x*, *t* , (18)

*δ*ð Þ*t*

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

*<sup>S</sup>* ð Þ *x* is a characteristic function of set *S*�, which is equal to 0.5 on its

*D <sup>∂</sup><sup>t</sup>* ¼ �*<sup>n</sup> jH*�

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

Consider the action of the KG-operator on *u*^. Since ½ � *u <sup>S</sup>* ¼ �*u*, and performing

*<sup>δ</sup>S*ð Þ *<sup>x</sup> H t*ðÞ� *H t*ð Þ *unjδS*ð Þ *<sup>x</sup>* � �, *<sup>j</sup>* � *<sup>c</sup>*

where *δS*ð Þ *x H t*ð Þ is simple layer on lateral surface of a space-time cylinder

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)

*<sup>D</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* , ^*<sup>f</sup>* <sup>¼</sup> *f x*ð Þ , *<sup>t</sup> <sup>H</sup>*�

�2 *H*�

*<sup>S</sup>* ð Þ *<sup>x</sup> <sup>u</sup>*\_ <sup>0</sup>ð Þ *<sup>x</sup> <sup>δ</sup>*ð Þþ*<sup>t</sup>* ^*f x*ð Þ *; <sup>t</sup>* , (19)

the zero solution. For him, f = 0, the initial conditions and the corresponding

2 *q x*ð Þ*u*<sup>2</sup> <sup>ð</sup>*x*, *<sup>t</sup>*Þg *dV x*ð Þ¼ <sup>0</sup>*:* �

boundary conditions are also zero. Then from Theorem 1, it follows:

*E x*ð Þþ , *t* 0, 5*c*

Consider the case when scattering potential is constant:

*H*�

do this, we introduce the characteristic function of the solution domain

*<sup>D</sup>*ð Þ� *x*, *t H*�

boundary *S*; and *H t*ð Þ is Heaviside's function, which is equal to 0.5 at *t* = 0. *H*�

characteristic function of space-time cylinder *D*�. It is easy to show that:

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

**Proof.** Due to the linearity of the problem, it suffices to prove the uniqueness of

*Gordon equation is unique.*

proved.

**potential**

where *H*�

□*cu*^ � *<sup>m</sup>*<sup>2</sup>

*D* ¼ *S* � *R*<sup>þ</sup> f g.

**43**

ð

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

*S*�

*∂H*� *D ∂x <sup>j</sup>*

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

this, we put regular generalized functions:

generalized differentiation using (17), we get

*<sup>u</sup>*^ ¼ � *<sup>∂</sup><sup>u</sup> ∂n*

> � *c* �2 *H*�

are known, and the given initial conditions.

*u*^ ¼ *u x*ð Þ , *t H*�

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

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

$$\left|\boldsymbol{u}\_{\mathcal{S}}(\boldsymbol{x},t) - \boldsymbol{u}\_{\mathcal{S}}(\boldsymbol{y},t)\right| \leq const \left\|\boldsymbol{x} - \boldsymbol{y}\right\|^{\beta},\tag{16}$$

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

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

$$\begin{aligned} &\int\_{S^{-}} (E(\mathbf{x},t) - E\_0(\mathbf{x})) \, \, dV(\mathbf{x}) + \mathbf{0}.5c^2 \int\_{S^{-}} q(\mathbf{x}) \Big( u^2(\mathbf{x},t) - u\_0^2(\mathbf{x}) \Big) \, \, dV(\mathbf{x}) \\ &= c^2 \int\_0^t dt \int\_{S^{-}} (\dot{u}s(\mathbf{x},t)p(\mathbf{x},t))dS(\mathbf{x}) - c^2 \int\_0^t dt \int\_{S^{-}} (\dot{r}(\mathbf{x},t)\dot{u}(\mathbf{x},t) \, \, dV(\mathbf{x}) \end{aligned}$$

**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> , which is continuous in the regions between the fronts. Therefore, using the Ostrogradsky-Gauss theorem in *R<sup>N</sup>*þ<sup>1</sup> , we get

$$\begin{aligned} &\int\_{D^{-}} \left(c^{-2}E + \frac{1}{2}q(\mathbf{x})u^{2}\right)\_{,t}dV(\mathbf{x},t) - \int\_{D^{-}} (\dot{\mathbf{u}}\,u\_{,i})\_{,j}dV(\mathbf{x},t) \\ &+ \int\_{D^{-}} \dot{u}f(\mathbf{x},t)dV(\mathbf{x},t) = \int\_{D^{-}} \dot{u}f(\mathbf{x},t)dV(\mathbf{x},t) + \\ &+ \int\_{S^{-}} \left\{c^{-2}(E(\mathbf{x},t) - E\_{0}(\mathbf{x})) + \frac{1}{2}q(\mathbf{x})\left(u^{2}(\mathbf{x},t) - u\_{0}^{2}(\mathbf{x})\right)\right\}dV(\mathbf{x}) - \\ &- \int\_{0}^{t} \left(\dot{u}\frac{\partial u}{\partial n}\right)dS(\mathbf{x})dt + \sum\_{F\_{k}}\int\_{F\_{k}} \left[c^{-2}E\nu\_{t} - \frac{\partial u}{\partial \nu}\dot{u}\right]\_{F\_{k}}dF\_{k}(\mathbf{x},t) = 0 \end{aligned}$$

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 virtue of (3) and Lemma 2,

$$\left[c^{-2}E\nu\_t - \dot{u}\mu\_{,j}\nu\_j\right]\_{F\_k} = -||\nu||\_N c^{-1} \left[E + c\dot{u}\frac{\partial u}{\partial n}\right] = \mathbf{0}.$$

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 the Theorem 2.

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

**Theorem 2.** If *q x*ð Þ≥0*, then the classic solution of first (second) BVP for Klein-Gordon equation is unique.*

**Proof.** Due to the linearity of the problem, it suffices to prove the uniqueness of the zero solution. For him, f = 0, the initial conditions and the corresponding boundary conditions are also zero. Then from Theorem 1, it follows:

$$\int\_{\mathcal{S}^-} \left\{ E(\varkappa, t) + 0, \mathfrak{K}^2 q(\varkappa) u^2(\varkappa, t) \right\} \, dV(\varkappa) = \mathbf{0}.$$

Since both terms are nonnegative, therefore, *E* = 0 and *u* = 0. The theorem is proved.
