**1. Introduction**

Klein-Gordon equation is hyperbolic differential equation in partial derivatives of second order. As is known, a class of solutions of hyperbolic equations contains no differentiable functions that have discontinuous derivatives on characteristic surfaces. By these cause, the construction of solutions for such equations, using smoothness of their differentiability, is impossible. However, this type of solution describes the shock waves, a presence of which is typical for physical processes.

Fundamental solutions of hyperbolic equations are singular generalized functions, the features of which are concentrated on moving surfaces—wave fronts, which propagate at a certain speed (it is called *light* or *sound speed*). This sharply distinguishes them from the fundamental solutions of elliptic and parabolic equations that are singular at a point. Therefore, classical methods of constructing boundary integral equations for solving nonstationary boundary value problems (BVP) based on the methods of potential theory or Green's formulas are unsuitable. To solve nonstationary boundary value problems for hyperbolic equations based on these methods, the Laplace or Fourier transform is usually used, which leads hyperbolic equations to parameterized elliptic type equations in the spaces of transforms. They are solved on the basis of the direct or indirect method of boundary integral equations (BIE). To restore original solutions, various numerical

procedures of the inverse Fourier transform, Laplace, and others are used. The class of such problems of mathematical physics began to be considered as early as the 1970s of the last century, which was connected with the advent of computing technology, but so far, the number of works in this direction is very limited. As is known, the inverse transformation procedures are unstable, which are typical for solutions of the Fredholm equations of the first type and require regularization of the corresponding equations to obtain reliable results. Therefore, the problem of developing constructive methods for solving initial-boundary value problems for hyperbolic equations and systems of equations of mathematical physics for studying of various wave processes in bodies and in continuous medium remains relevant to the present time.

*ν*2 *<sup>t</sup>* � *c*

, *<sup>x</sup>*∈*RN*, ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∈</sup>*RN*þ<sup>1</sup>

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

of *F* at fixed *t*), which moves with speed *c*:

If solution of (1) is continuous:

convolution). Such solutions (1) are called *shock waves*.

and directed forward its propagation. It is obvious that,

fronts. If before the wave front *u* � 0, then at the wave front:

ð Þ¼� *grad u*, *n c*

corresponding partial derivative: *<sup>u</sup>*, *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup><sup>u</sup>*

^*f x*ð Þ , *<sup>t</sup>* , by considering it as a generalized.

matical physics problems.

**39**

*F* in *RN*þ<sup>1</sup>

jumps:

2X *N*

*j*¼1 *ν*2

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

where *ν*ð Þ¼ *x*, *t* ð Þ *ν*1, … , *νN*,*ν<sup>t</sup>* is the normal vector to the characteristic surface

—the light cone at *ν<sup>t</sup>* <0. The solution of equation of *F*(1) and its derivatives can be discontinuous. In *RN*, characteristic surface *F* corresponds to wave front *Ft* (section

*c* ¼ �*νt=*k k*ν <sup>N</sup>*, k k*ν <sup>N</sup>* ¼ ffiffiffiffiffiffiffiffi

then wave fronts of Hadamard's conditions of continuity are satisfied under

*Ft*

for *x*∈*Ft*, where *n x*ð Þ , *t* is wave vector. It is a unit vector, normal to wave front *Ft*

Hereinafter, for abbreviation of the record, a symbol after comma defines the

*∂x <sup>j</sup>*

the continuity condition (4) and ensures continuity on the *Ft* of tangent derivatives of *u*. The condition (5) is the law of conservation of momentum at the shock wave

�1

To study the solutions of the KG-Eq, it is convenient to use the apparatus of the theory of generalized functions, which allows one to investigate shock waves, as well as singular solutions from the class of generalized functions typical for mathe-

For this purpose, let us consider KG-Eq (1) on the space of generalized functions *D*<sup>0</sup> *R<sup>N</sup>*þ<sup>1</sup> � �, which is the space of linear continuous functionals on the space of basic functions *D R<sup>N</sup>*þ<sup>1</sup> � �, which are finite infinitely differentiable functions [1]. Further, the usual locally integrable function (*regular*) *f x*ð Þ , *t* will be marked from the top

If *u* (*x,t*) is regular differentiable and has a finite discontinuity on *F*, then in

*D*<sup>0</sup> *R<sup>N</sup>*þ<sup>1</sup> � �, as is known [1], its partial derivative is equal to the following:

*u*\_, *x*∈*Ft:*

*u*^, *<sup>j</sup>* ¼ *u*, *<sup>j</sup>* þ ½ � *u <sup>F</sup>νjδF*, (8)

*u*\_ þ *cn <sup>j</sup> u*, *<sup>j</sup>* � �

*un*\_ *<sup>j</sup>* þ *cu*, *<sup>j</sup>* � �

(here and further throughout in order to reduce the record on repeated indexes in the product, the summation from 1 to *N* is carried out, which is similar to tensor

*<sup>j</sup>* ¼ 0, (2)

p (3)

. It corresponds to the cone of characteristic normals

*ν <sup>j</sup>ν <sup>j</sup>*

½ � *u x*ð Þ , *t Ft* ¼ 0, (4)

*Ft* ¼ 0, *j* ¼ 1, *N*; (5)

. The condition (5) is a consequence of

*ni* ¼ *νi=*k k*ν <sup>N</sup>*, *i* ¼ 1, *N*; (7)

¼ 0, (6)

An effective method for solving boundary value problems for hyperbolic equations and systems of mixed type is the method of generalized functions (MGF), which allows to move from solving boundary value problems to solving the corresponding differential equations in the space of generalized functions and build integral representations of generalized solutions of initial boundary value problems in the original space-time to investigate the processes accompanied by shock waves.

To solve the Cauchy problem for hyperbolic equations, this method was proposed by Vladimirov [1]. In Refs. [2–6], the MGF was developed to solve nonstationary and stationary boundary value problems of the theory of elasticity, thermoelasticity, and electrodynamics and initial-boundary value problems for hyperbolic equations systems which are typical to the mathematical physics [7].

In the present chapter, this method is used to solve initial-boundary value problems for the Klein-Gordon equation (KG-Eq), a hyperbolic equation of the theory of elementary particles of quantum mechanics [8]. Here, nonstationary boundary value problems for KG-Eq with Dirichlet or Neumann conditions on the boundary of the domain of definition are considered and the uniqueness of the stated boundary value problems taking into account of shock waves is proved. Based on the method of generalized functions, the boundary integral equations method (BIEM) was developed to solve the stated tasks. Dynamic analogs of Green's formulas for their solutions in the space of generalized functions are obtained and regular integral representations in the plane and three-dimensional cases are constructed. Solving singular boundary integral equations are obtained for solving the stated initial-boundary value problems.

## **2. Klein-Gordon equation: shock waves**

Klein-Gordon equation is formulated as:

$$
\Box\_{\epsilon}u + q(\varkappa)u = f(\varkappa, t). \tag{1}
$$

Here, we denote the wave operator,

$$
\Box\_{\mathcal{L}} = \Delta - c^{-2} \frac{\partial^2}{\partial t^2},
$$

where <sup>Δ</sup> <sup>¼</sup> <sup>P</sup>*<sup>N</sup> k*¼1 *∂*2 *∂x*<sup>2</sup> *k* is Laplace operator, *<sup>x</sup>*∈*R<sup>N</sup>*, *<sup>t</sup>*<sup>∈</sup> ½ Þ 0, <sup>∞</sup> , and scattering potential *q x*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *<sup>R</sup><sup>N</sup>* � �. It is a hyperbolic equation. Its characteristic equation has the next form:

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

$$
\omega\_t^2 - c^2 \sum\_{j=1}^N \nu\_j^2 = \mathbf{0},
\tag{2}
$$

where *ν*ð Þ¼ *x*, *t* ð Þ *ν*1, … , *νN*,*ν<sup>t</sup>* is the normal vector to the characteristic surface *F* in *RN*þ<sup>1</sup> , *<sup>x</sup>*∈*RN*, ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∈</sup>*RN*þ<sup>1</sup> . It corresponds to the cone of characteristic normals —the light cone at *ν<sup>t</sup>* <0. The solution of equation of *F*(1) and its derivatives can be discontinuous. In *RN*, characteristic surface *F* corresponds to wave front *Ft* (section of *F* at fixed *t*), which moves with speed *c*:

$$
\mathcal{L} = -\nu\_t / \|\nu\|\_{N^\bullet} \quad \|\nu\|\_{N} = \sqrt{\nu\_j \nu\_j} \tag{3}
$$

(here and further throughout in order to reduce the record on repeated indexes in the product, the summation from 1 to *N* is carried out, which is similar to tensor convolution). Such solutions (1) are called *shock waves*.

If solution of (1) is continuous:

procedures of the inverse Fourier transform, Laplace, and others are used. The class of such problems of mathematical physics began to be considered as early as the 1970s of the last century, which was connected with the advent of computing technology, but so far, the number of works in this direction is very limited. As is known, the inverse transformation procedures are unstable, which are typical for solutions of the Fredholm equations of the first type and require regularization of the corresponding equations to obtain reliable results. Therefore, the problem of developing constructive methods for solving initial-boundary value problems for hyperbolic equations and systems of equations of mathematical physics for studying of various wave processes in bodies and in continuous medium remains relevant to

*Mathematical Theorems - Boundary Value Problems and Approximations*

An effective method for solving boundary value problems for hyperbolic equations and systems of mixed type is the method of generalized functions (MGF), which allows to move from solving boundary value problems to solving the corresponding differential equations in the space of generalized functions and build integral representations of generalized solutions of initial boundary value problems in the original space-time to investigate the processes accompanied by

To solve the Cauchy problem for hyperbolic equations, this method was pro-

nonstationary and stationary boundary value problems of the theory of elasticity, thermoelasticity, and electrodynamics and initial-boundary value problems for hyperbolic equations systems which are typical to the mathematical physics [7]. In the present chapter, this method is used to solve initial-boundary value problems for the Klein-Gordon equation (KG-Eq), a hyperbolic equation of the theory of elementary particles of quantum mechanics [8]. Here, nonstationary boundary value problems for KG-Eq with Dirichlet or Neumann conditions on the boundary of the domain of definition are considered and the uniqueness of the stated boundary value problems taking into account of shock waves is proved. Based on the method of generalized functions, the boundary integral equations method (BIEM) was developed to solve the stated tasks. Dynamic analogs of Green's formulas for their solutions in the space of generalized functions are obtained and regular integral representations in the plane and three-dimensional cases are constructed. Solving singular boundary integral equations are obtained for

□*<sup>c</sup>* <sup>¼</sup> <sup>Δ</sup> � *<sup>с</sup>*�<sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

potential *q x*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *<sup>R</sup><sup>N</sup>* � �. It is a hyperbolic equation. Its characteristic equation has

*∂t*2 ,

is Laplace operator, *<sup>x</sup>*∈*R<sup>N</sup>*, *<sup>t</sup>*<sup>∈</sup> ½ Þ 0, <sup>∞</sup> , and scattering

□*cu* <sup>þ</sup> *q x*ð Þ*<sup>u</sup>* <sup>¼</sup> *f x*ð Þ , *<sup>t</sup> :* (1)

posed by Vladimirov [1]. In Refs. [2–6], the MGF was developed to solve

solving the stated initial-boundary value problems.

**2. Klein-Gordon equation: shock waves**

Klein-Gordon equation is formulated as:

Here, we denote the wave operator,

*k*¼1 *∂*2 *∂x*<sup>2</sup> *k*

where <sup>Δ</sup> <sup>¼</sup> <sup>P</sup>*<sup>N</sup>*

the next form:

**38**

the present time.

shock waves.

$$[\mathfrak{u}(\mathfrak{x},t)]\_{F\_t} = \mathbf{0},\tag{4}$$

then wave fronts of Hadamard's conditions of continuity are satisfied under jumps:

$$\left[\dot{u}n\_{j} + cu\_{j}\right]\_{F\_{t}} = 0, \quad j = \overline{1, N};\tag{5}$$

$$\left[\dot{u} + c n\_j u, \iota\_j\right]\_{F\_t} = \mathbf{0},\tag{6}$$

for *x*∈*Ft*, where *n x*ð Þ , *t* is wave vector. It is a unit vector, normal to wave front *Ft* and directed forward its propagation. It is obvious that,

$$m\_i = \nu\_i / \|\nu\|\_N, \quad i = \mathbf{1}, N; \tag{7}$$

Hereinafter, for abbreviation of the record, a symbol after comma defines the corresponding partial derivative: *<sup>u</sup>*, *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup><sup>u</sup> ∂x <sup>j</sup>* . The condition (5) is a consequence of the continuity condition (4) and ensures continuity on the *Ft* of tangent derivatives of *u*. The condition (5) is the law of conservation of momentum at the shock wave fronts. If before the wave front *u* � 0, then at the wave front:

$$(\operatorname{grad} u, n) = -c^{-1}\dot{u}, \quad x \in F\_t.$$

To study the solutions of the KG-Eq, it is convenient to use the apparatus of the theory of generalized functions, which allows one to investigate shock waves, as well as singular solutions from the class of generalized functions typical for mathematical physics problems.

For this purpose, let us consider KG-Eq (1) on the space of generalized functions *D*<sup>0</sup> *R<sup>N</sup>*þ<sup>1</sup> � �, which is the space of linear continuous functionals on the space of basic functions *D R<sup>N</sup>*þ<sup>1</sup> � �, which are finite infinitely differentiable functions [1]. Further, the usual locally integrable function (*regular*) *f x*ð Þ , *t* will be marked from the top ^*f x*ð Þ , *<sup>t</sup>* , by considering it as a generalized.

If *u* (*x,t*) is regular differentiable and has a finite discontinuity on *F*, then in *D*<sup>0</sup> *R<sup>N</sup>*þ<sup>1</sup> � �, as is known [1], its partial derivative is equal to the following:

$$
\hat{\boldsymbol{\mu}}\_{\cdot j} = \boldsymbol{\mu}\_{\cdot j} + [\boldsymbol{\mu}]\_{\boldsymbol{F}} \boldsymbol{\nu}\_{\circ} \delta\_{\boldsymbol{F}},\tag{8}
$$

where the first term on the right is the classical derivative of *x <sup>j</sup>*, *δF*(*x,t*) is a simple layer on *F*, a singular generalized function [1], k k**ν** ¼ 1. Using (8), it is possible to determine the second derivatives sequentially.

*Definition*. A solution *u*(*x, t*) of Eq (1), which is continuous together with derivatives up to the second order almost everywhere, with the exception of a finite or countable number of discontinuity surfaces (wave fronts), on which the conditions (5) and (6) for jumps are satisfied, is called as classical solution.

Here, *n* is the normal to the shock wave front in *RN*. This implies the first

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

*E*, *<sup>t</sup>* � *u u*\_ , *<sup>j</sup>*

½ �þ *E c u u*\_ , *<sup>j</sup>*

Similarly, we can get the equation for *L* by multiplying Eq. (1) by *u*:

 , *<sup>j</sup>* <sup>¼</sup> *u c*�<sup>2</sup> ½ � *<sup>u</sup>*\_ *<sup>ν</sup><sup>t</sup>* � *<sup>u</sup>*, *<sup>j</sup>*

It means that the Lagrange function is continuous at the shock wave fronts.

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

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*;

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

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

*D* ¼ *S* � *R*<sup>þ</sup> f g is lateral surface of space-time cylinder *D*� ¼ *S*� � *R*þ,

ð Þ BVP II *<sup>∂</sup><sup>u</sup>*

and boundary data on the velocities is not satisfied

*<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>*

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

ð Þ *uu*\_ , *<sup>t</sup>* � *uu*, *<sup>j</sup>*

Taking into account (3) and (4), for shock waves, we get

ð Þ *uu*\_ , *<sup>t</sup>* � *uu*, *<sup>j</sup>*

field of differentiability of solutions, we have the equation:

*c* �2

, *<sup>j</sup>* <sup>þ</sup> *u f* \_ <sup>¼</sup> *<sup>c</sup>*

For the right side to be turned to zero, it is necessary

The latter coincides with the formula of Lemma 2.

*L* þ *c* �2

�2

�½ �¼ *L c*

**conservation law**

Neumann conditions:

**41**

For shock waves in *D*<sup>0</sup> *RN*þ<sup>1</sup> , it has the form:

*<sup>E</sup>*\_ � *u u*\_ , *<sup>j</sup>*

The condition for a jump in the energy density at the shock wave front can be obtained more easily by considering the corresponding energy equation in *D*<sup>0</sup> *RN*þ<sup>1</sup> , which we get by multiplying the Eq. (1) by *u*\_ . After simple transformations in the

�2

¼ �k k*ν <sup>N</sup> c*

*nj* <sup>¼</sup> <sup>0</sup>*:*

½ � *E ν<sup>t</sup>* � *u u*\_ , *<sup>j</sup>*

�1

 *ν<sup>j</sup> δ<sup>F</sup>* <sup>¼</sup>

> ½ �þ *E u u*\_ , *<sup>j</sup> nj δFt*

, *<sup>j</sup>* <sup>þ</sup> *u f* \_ <sup>¼</sup> <sup>0</sup> (9)

, *<sup>j</sup>* <sup>þ</sup> *q u*<sup>2</sup> � *u f* <sup>¼</sup> <sup>0</sup> (10)

 *ν<sup>j</sup>* <sup>¼</sup> <sup>0</sup>

*<sup>∂</sup><sup>n</sup>* ¼ *u*, *jn <sup>j</sup>*.

*u x*ð Þ¼ , 0 *u*0ð Þ *x* for *x*∈ *S*� þ *S* (11) *u x* \_ð Þ¼ , 0 *v*0ð Þ *x* for *x*∈*S*� (12)

ð Þ BVP I *u x*ð Þ¼ , *t uS*ð Þ *x*, *t* for *x*∈*S* (13)

*<sup>∂</sup><sup>n</sup>* <sup>¼</sup> *p x*ð Þ , *<sup>t</sup>* for *<sup>x</sup>*∈*<sup>S</sup>* (14)

formula of the lemma.

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

*c* �2

**Lemma 1.** *If u x*ð Þ , *t is classic solution of* (1), *then u x* ^ð Þ , *t is generalized solution of it*. **Proof.** Taking into account these equations and (3), we get

ð Þ □*<sup>c</sup>* <sup>þ</sup> *q x*ð Þ *<sup>u</sup>*^ <sup>¼</sup> *f x*ð Þþ *; <sup>t</sup> <sup>c</sup>* �<sup>1</sup> *<sup>u</sup>*, *<sup>t</sup>* ½ �*Ft* <sup>þ</sup> *nju*, *<sup>j</sup>* � � *Ft* n ok k*<sup>ν</sup> <sup>N</sup>δ<sup>F</sup>* <sup>þ</sup> *<sup>c</sup>* �1 *<sup>∂</sup><sup>t</sup>* k k*<sup>ν</sup> <sup>N</sup>*½ � *<sup>u</sup> Ft δF*Þ n o <sup>þ</sup> *<sup>∂</sup><sup>j</sup>* k k*<sup>ν</sup> <sup>N</sup>*½ � *<sup>u</sup> Ft njδ<sup>F</sup>* n o

By virtue of (4) and (6), the densities of simple and double layers here are equal to zero on the right, which were required to be proved.

From this lemma, it follows that the conditions on the fronts of shock waves are easy to obtain, considering the classical solutions of hyperbolic equations as generalized. It is enough to equate to zero the density of the corresponding independent singular generalized functions—analogs of simple, double, and other layers arising from the generalized differentiation of solutions. The determination of such conditions on the basis of classical methods is a very time-consuming procedure.

Let us put the energy density of the *u*-field *E* and the Lagrange *L* function:

$$E = 0.5 \left( \dot{u}^2 + c^2 \sum\_{j=1}^{N} u\_{,j}^2 \right),$$

$$L = 0.5 \left( \dot{u}^2 - c^2 \sum\_{j=1}^{N} u\_{,j}^2 \right).$$

**Lemma 2.** *If the classical solution of the KG-Eq.* (1), *then the following conditions for energy density jumps and Lagrange functions are satisfied at the shock wave fronts:*

$$[E]\_{F\_t} = -c \left[ \dot{u} \frac{\partial u}{\partial n} \right]\_{F\_t},$$

$$[L(\mathbf{x}, t)]\_{F\_t} = \mathbf{0}.$$

**Proof:** It is easy to show that for jumps, the equation is fulfilled:

$$[ab] = a^+[b] + b^-[a],$$

where the plus and minus signs indicate the limiting values of the functions *a* and *b* on the wave front from the side of the wave vector and opposite. Using this equality and the Hadamard's conditions (4) and (5), we get

$$\left[E + c\dot{u}\frac{\partial u}{\partial n}\right] = c^2 \left[0.5\left(c^{-1}\dot{u}^2 + cu\_j u\_j\right) + \dot{u}\frac{\partial u}{\partial n}\right] = \dots =$$

$$= 0.5c[\dot{u}]\left(\dot{u}^- + cu\_j^- n\_j\right) + 0.5c^2[u\_j]\left(cu\_j^- + \dot{u}^- n\_j\right) = 0$$

$$= 0.5c^3u\_j^-\left[u\_{,j} + c^{-1}n\_j\dot{u}\right] + 0.5c\dot{u}^-\left[c n\_j u\_{,j} + \dot{u}\right] = 0$$

**40**

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

Here, *n* is the normal to the shock wave front in *RN*. This implies the first formula of the lemma.

The condition for a jump in the energy density at the shock wave front can be obtained more easily by considering the corresponding energy equation in *D*<sup>0</sup> *RN*þ<sup>1</sup> , which we get by multiplying the Eq. (1) by *u*\_ . After simple transformations in the field of differentiability of solutions, we have the equation:

$$
\varepsilon^{-2}E\_{,t} - (\dot{u}\,u\_{,j})\_{,j} + \dot{u}f = 0 \tag{9}
$$

For shock waves in *D*<sup>0</sup> *RN*þ<sup>1</sup> , it has the form:

where the first term on the right is the classical derivative of *x <sup>j</sup>*, *δF*(*x,t*) is a simple layer on *F*, a singular generalized function [1], k k**ν** ¼ 1. Using (8), it is

*Definition*. A solution *u*(*x, t*) of Eq (1), which is continuous together with derivatives up to the second order almost everywhere, with the exception of a finite or countable number of discontinuity surfaces (wave fronts), on which the condi-

�<sup>1</sup> *<sup>u</sup>*, *<sup>t</sup>* ½ �*Ft* <sup>þ</sup> *nju*, *<sup>j</sup>*

*njδ<sup>F</sup>* n o

tions on the basis of classical methods is a very time-consuming procedure. Let us put the energy density of the *u*-field *E* and the Lagrange *L* function:

*E* ¼ 0*:*5 *u*\_

*L* ¼ 0*:*5 *u*\_

n o

**Lemma 1.** *If u x*ð Þ , *t is classic solution of* (1), *then u x* ^ð Þ , *t is generalized solution of it*.

� � *Ft*

By virtue of (4) and (6), the densities of simple and double layers here are equal

From this lemma, it follows that the conditions on the fronts of shock waves are easy to obtain, considering the classical solutions of hyperbolic equations as generalized. It is enough to equate to zero the density of the corresponding independent singular generalized functions—analogs of simple, double, and other layers arising from the generalized differentiation of solutions. The determination of such condi-

<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*

<sup>2</sup> � *<sup>c</sup>*

**Lemma 2.** *If the classical solution of the KG-Eq.* (1), *then the following conditions for*

*energy density jumps and Lagrange functions are satisfied at the shock wave fronts:*

½ � *E Ft* ¼ �*c u*\_

**Proof:** It is easy to show that for jumps, the equation is fulfilled:

equality and the Hadamard's conditions (4) and (5), we get

<sup>2</sup> 0*:*5 *c* �1 *u*\_

¼ 0*:*5*c*½ � *u*\_ *u*\_

¼ 0*:*5*c* 3 *u*, � *<sup>j</sup> u*, *<sup>j</sup>* þ *c*

*E* þ *cu*\_

**40**

� �

*∂u ∂n*

¼ *c*

½ � *L x*ð Þ , *t Ft* ¼ 0*:*

½ �¼ *ab a*þ½ �þ *b b*�½ � *a* ,

where the plus and minus signs indicate the limiting values of the functions *a* and *b* on the wave front from the side of the wave vector and opposite. Using this

> <sup>2</sup> <sup>þ</sup> *cu*, *ju*, *<sup>j</sup>* � � <sup>þ</sup> *<sup>u</sup>*\_

> > �1 *nju*\_ � � <sup>þ</sup> <sup>0</sup>*:*5*cu*\_

� þ *cu*, � *<sup>j</sup> nj* � �

� �

2X *N*

!

*j*¼1 *u*, 2 *j*

2X *N*

!

*j*¼1 *u*, 2 *j*

*∂u ∂n* � �

*Ft* ,

> *∂u ∂n*

> > <sup>2</sup> *u*, *<sup>j</sup>* � � *cu*,

þ 0*:*5*c*

¼ … ¼

� *<sup>j</sup>* þ *u*\_ �*nj* � �

� *cnju*, *<sup>j</sup>* <sup>þ</sup> *<sup>u</sup>*\_ � � <sup>¼</sup> <sup>0</sup>

¼

,

*:*

k k*ν <sup>N</sup>δ<sup>F</sup>* þ *c*

�1

*<sup>∂</sup><sup>t</sup>* k k*<sup>ν</sup> <sup>N</sup>*½ � *<sup>u</sup> Ft*

n o

*δF*Þ

tions (5) and (6) for jumps are satisfied, is called as classical solution.

**Proof.** Taking into account these equations and (3), we get

possible to determine the second derivatives sequentially.

*Mathematical Theorems - Boundary Value Problems and Approximations*

<sup>þ</sup> *<sup>∂</sup><sup>j</sup>* k k*<sup>ν</sup> <sup>N</sup>*½ � *<sup>u</sup> Ft*

to zero on the right, which were required to be proved.

ð Þ □*<sup>c</sup>* <sup>þ</sup> *q x*ð Þ *<sup>u</sup>*^ <sup>¼</sup> *f x*ð Þþ *; <sup>t</sup> <sup>c</sup>*

$$\begin{aligned} \left\{ \boldsymbol{\omega}^{-2} \dot{\boldsymbol{E}} - \left( \dot{\boldsymbol{u}} \,\boldsymbol{u},\_{\boldsymbol{j}} \right) , \boldsymbol{j} + \dot{\boldsymbol{u}} \boldsymbol{f} = \left\{ \boldsymbol{\omega}^{-2} [\boldsymbol{E}] \boldsymbol{\nu}\_{l} - \left[ \dot{\boldsymbol{u}} \,\boldsymbol{u},\_{\boldsymbol{j}} \right] \boldsymbol{\nu}\_{\boldsymbol{j}} \right\} \delta\_{\boldsymbol{F}} = \\ &= -||\boldsymbol{\nu}||\_{N} \left\{ \boldsymbol{\omega}^{-1} [\boldsymbol{E}] + \left[ \dot{\boldsymbol{u}} \,\boldsymbol{u},\_{\boldsymbol{j}} \right] \boldsymbol{n}\_{\boldsymbol{j}} \right\} \delta\_{\boldsymbol{F}, \boldsymbol{u}} \end{aligned}$$

For the right side to be turned to zero, it is necessary

$$[E] + c \left[ \dot{u} \, u\_{\ast j} \right] \eta\_j = 0.$$

The latter coincides with the formula of Lemma 2. Similarly, we can get the equation for *L* by multiplying Eq. (1) by *u*:

$$L + c^{-2}(u\dot{u})\_{,t} - (uu\_{,j})\_{,j} + qu^2 - uf = \mathbf{0} \tag{10}$$

Taking into account (3) and (4), for shock waves, we get

$$-[L] = \left[\mathfrak{c}^{-2}(\mathfrak{uu})\_t - (\mathfrak{uu}\_{,j})\_{,j}\right] = \mathfrak{u}\left\{\mathfrak{c}^{-2}[\dot{\mathfrak{u}}]\nu\_t - [\mathfrak{u}\_{,j}]\nu\_{\dot{\mathfrak{f}}}\right\} = \mathbf{0}$$

It means that the Lagrange function is continuous at the shock wave fronts.
