BOUNDARY INTEGRAL EQUATIONS METHOD AT EDGE PROBLEMS FOR KLEIN–GORDONFOX EQUATIONS
Alexeeva L.A., Baegizova A.S.
KR Science & Education Ministry, Mathematics Institute, alexeeva47@mail.ru,
L.N. Gumilev Eurasian National University, baegiz60@mail.ru
Non-stationary boundary problems for quantum mechanics equation named Klein-GordonFock equation under Dirichlet or Neumann conditions at the boundary of definition domain are
considered, uniqueness of the set of edge problems with shock waves is proved. On the basis of
the method of generalized functions the method of boundary integral equations for solving them
is designed. The dynamic analogs of Green formulas in the space of generalized functions are
obtained, and the regular integral representations in the plane and three-dimensional cases are
built. Resoluting singular boundary integral equations for solving the initial-edge problems are
obtained.
1. Klein-Gordon-Fock equations. Shock waves
Klein-Gordon-Fock equation is formulated as
cu
 q( x)u  u  с
2
 u
2
t
2
 q( x)u  f ( x, t ),
(1.1)
Here x  R N , t [0, ) , q( x) - potential of dispersion. This equation has hyperbolic shape. Its
characteristic equation is
νt2
c
2
N

j 1

ν 2j
 0,
where  ( x, t )   1 ,..., N , t
c  νt / 
,
N

N

N
 ν2j
j 1
,
t  0
 is normal vector to characteristic surface in
(1.2)
R N 1  {(x, t )} ,
that
defined by F. Corresponding cone of characteristic normals is light cone at  t  0 .
In R N characteristic surface F corresponds to wave front Ft (section of F at fixed t),
which moves with speed c. Those solutions of (1.1) are named shock waves. If solution of (1.1)
is continuous:
u  x, t   F  0
t
(1.3)
then on wave fronts Adamar’s conditions of continuity are satisfied under jumps:
un j  cu, j   0 , u  cn j u, j   0,
Ft
Ft
j  1, N ;
for xFt
(1.4)
1
where n(x,t) – wave vector – is unit normal vector to Ft directed forward propagation of wave
front: ni   i /  N , i  1, N .
Lemma 1.1. If u is classic solution of (1.1), then û is generalized solution of (1.1).
Proof:


( c  q( x))uˆ  f ( x, t ) H (t )  c -1 u,t F   n j u, j  
Ft
t

c 1 t 
N


[u ]Ft  F )   j 
N

N
F 
[u ]Ft n j F  fˆ ( x, t )
Consequence. It is easy to take the conditions on fronts of shock waves considering the
classic solutions of hyperbolic equations as generalized. It is sufficiently to equal zero the densities of corresponding independent singular generalized functions – analogs of simple, double
and other layers, which arising at generalized differentiating of solutions. Definition of those
conditions on basis of classic methods is very hard work procedure.
2. Setup of non-stationary edge problems for Klein-Gordon-Fock equation. Energy
conservation law.
Let in region S   R N , bounded surface S, it constructs solution u(x,t) of (1.1) at t  0. Let’s introduce next marks: n( x) - vector of external normal to S, D  {S  R } - lateral surface of spacetime cylinder
D  S   R  , R   (0, ) ,
Initial conditions: At
t0
for
xS :
u ( x,0)  u0 ( x) ,
u
 u, j n j
n
- derivative of u on normal n at S.
u ( x,0)  v0 ( x) u0 ( x)  C ( S   S ) , v0 ( x)  L1 (S  ) ,
(2.1)
( 2.2)
Boundary problems corresponding to Dirichlet or Neumann conditions:
(BP I)
(BP II)
u ( x , t )  u S ( x, t )
u
 p( x, t ) ,
n
for
x  S , u S ( x, t ) -
p ( x, t )  L1 ( D )
for
Gelder’s on S,
xS
.
(2.3)
(2.4)
On fronts of shock waves Adamar’s conditions (1.4)-(1.6) are satisfied under jumps.
Lemma 2.1. If u is classic solution of Klein-Gordon-Fock equation (1.1), then on fronts of
shock waves next conditions (1.4)-(1.6) are satisfied under jumps:
 E Ft
 2 2 N 2
 u 
 0,5 u,t c  u, j   c u 
 n  Ft

 Ft
j 1
2
 L( x, t )Ft
 2 2 N 2
 
u  
 0,5 u  c  u , j    u  с
  u Ft .

n

 Ft 
j 1

Theorem 2.1. (Energy conservation law). If u(x,t) is classic solution of edge problem, then
  E ( x, t )  E0 ( x) dV ( x)  0,5c  q( x) u
2
S

S
 c
 dt 
0
D

( x, t )  u02 ( x) dV ( x) 

t
2
2
t
f ( x, t )u,t dV ( x)  c

2
 uS ( x, t ) p( x, t )  dS ( x)dt
0S
Consequence. If q( x)  0 , then the classic solution of first (second) edge problem for KleinGordon-Fock equation is unique.
3. Dynamic analog of Green’s formula at q( x)   m
2
Let introduce regular generalized functions:
uˆ  u ( x, t ) H D ( x, t ),
fˆ  f ( x, t ) H D ( x, t ) ,
(3.0)
where u( x, t ) is classic solution of boundary problem H D ( x, t )  H S  x  H (t ) , where


H S  x  is characteristic function of set S  , it’s equal 0,5 on its boundary S,
side’s function, it’s equal 0,5 at t=0.
is easy to show, that:
H D
is Heavi-
is characteristic function of space-time cylinder
H D
  n j S ( x) H (t ),
x j
Where
H (t )
H D
 n j H S ( x) (t )
t
D .
It
(3.1)
 (t ) is Dirac’s function. In case of generalized functions we take:
u
 S  x  H  t   H  t  un j S  x  , j 
n
c 2 H S  x  u0  x   (t )  c 2 H S  x  u0  x   (t )  fˆ ( x, t ),
c uˆ
 m 2uˆ  

Where   x, t  S ( x) H (t ) is simple layer on lateral surface of cylinder


D  S  R
(3.2)
:
3
u
uˆ  u  x, t  H S  x  H  t   Uˆ   S  x  H  t   Uˆ  un j S  x  H  t  , j 
n
(3.3)
1 ˆ
 2 U  H S  x  u0  x  ,t c 2Uˆ  H S  x  u0  x   fˆ ( x, t )  Uˆ
x
x
c




Where Uˆ  x, t  is fundamental solution (Green’s function of equation (1.1)):
ˆ  m 2Uˆ   ( x) (t ) ,
cU
Uˆ  0 at
t  0,
(3.4)
Uˆ  0 at x  ct
(radiation conditions)
(3.5)
Formula (3.4), (3.5) express of solution of edge problems through boundary means of
sought function and its derivative on normal to boundary, that is they are same to Green’s formula for Laplace’s equation. By the way, by virtue of peculiarities of fundamental solutions of
hyperbolic equations on the wave front, whose shape depends on dimension of space, their integral representation gives diverging integrals, having derivatives from fundamental solution.
To build regular integral representations let introduce prototype function by t:
Wˆ  Uˆ    x  H  t   Uˆ  H  t   tWˆ  Uˆ
(3.6)
t
Wˆ
Wˆ
Hˆ  x, n, t  
nj 
x j
n
(3.7)
Theorem 3.1. Generalized solution of edge problems satisfies to equation:
u
uˆ  Uˆ   S  x  H  t   Wˆ , j un j ( x) S  x  H  t   Wˆ , j u0  x  n j  x   S  x  
x
n
c 2Uˆ  H   x  u  x   c 2 Uˆ  H   x  u  x  ,  fˆ  Uˆ
x
S
0

x
S
0

t
From Theorem 3.1 it is consequent, that solution of the problem is entirely defined by initial
data, boundary means of normal derivative of function u  x, t  and its speed u  u,t   t u . By analog with representation of Laplace’s equation solution, these formulas may be called dynamical
analog of Green’s formula.
Formula of Theorem 3.1 permits at once to go to its integral writing without regularization
of under integral functions on fronts.
Then let’s consider representation of solution of edge problem for Klein-Gordon-Fock 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 consider consequently solutions of two edge problems:
4
1. Cochin’s problem at f ( x, t )  0 ;
2. Edge problem at zero initial conditions and f ( x, t )  0 .
By virtue of linearity of equations, solutions of set edge problems may be obtained by superposing of solutions that edge problems with correction of boundary conditions for second
problem with account of boundary meanings of Cochin’s problem solutions. Solution of Cochin’s problem for that equation has been early obtained by Vladimirov V.S. (see [2]). Here it
gives the solution of problem 2 at N=2 (planar problem).
4. Integral representation of solution of edge problems for
Klein-Gordon-Fock equations at N=2
In planar case at N=2 Green’s function is regular generalized function as shape:
2
ch  m c 2t 2  x 
H (ct  x ) 

Uˆ 
2
2
c 2t 2  x
2
cos  m c 2t 2  x 
H (ct  x )


Uˆ 
2
2
c 2t 2  x
at q( x)  m
at q( x)  m
2
2
(4.1)
(4.2)
with weak peculiarity on front x  ct :
Uˆ 
1
2 c t  x
2 2
2
at x  ct  0
(4.3)
Its carrier is light cone x  ct .
1
1
Wˆ 
d0 (r , t ) * H (t ) 
d1 (r , t ),
2
2 c
(4.4)
1
r
Hˆ ( x, t , n) 
d 2 (r , t ) , r  x ,
2 c
n
(4.5)
where, for example, at q( x)  m :
2
d0 (r , t )  H (ct  r )

ch m c 2t 2  r 2

c 2t 2  r 2
5
c 2t 2  r 2
d1 (r , t )  H (ct  r )

ch  mz 

0
z r

ch m c 2t 2  r 2
1
d 2 (r , t )   H (ct  r )  
ct
r

2
2
dz,
   H (ct  r)r
c 2t 2  r 2



0

ch  mz 
z2  r2

3
dz.
As against U, W and H are continuous on the front. Meanings of that functions on the front
r  ct , t  0 :
W r ct  0, H r ct  0,
(4.6)
At r  0 it takes place asymptotical representation:
U
ch(mct )
1 r
 o(r ), H  
 O(1).
2 ct
2 r n
(4.7)
Now let’s go to integral writing for N=2.
Theorem 4.1 (dynamical analog of Gauss’s formula). solution of initial - edge problem
for Klein-Gordon-Fock equation – equation (1.1) at N=2, f=0 and zero initial data may be
presented as: for x  S
2 u  x, t  H (t ) 


r
H
сt

r
dS
(
y
)
d
r
,

u
y
,
t









 d 
2

r 

n
(
y
)

St ( x )
t
c
c

t
H  сt  r  dS ( y )  d0  r ,  p  y, t    d ;
r
c
St ( x )

Here let’s introduce next marks: St


( x)  y  S  , r  ct , St ( x)   y  S , r  ct ,
r  yx
5. Singular boundary integral equations for planar initial – edge problems
Theorem 5.1. For x  S solution of edge problem satisfied to boundary integral equations as
next shape:
6
t
r
 u  x, t  H (t )  V .P.  H  сt  r 
dS ( y)  d 2  r ,  u  y, t    d 

n
(
y
)
S ( x)
r/c
t
c

St ( x )
H  сt  r  dS ( y )
t
 d0  r ,  p  y, t    d ,
r  yx
r /c
For first edge problem unknown derivative on normal enters under surface integral with
weak polar kernel. Other members are known. It is non classic equation, seems it’s nearer by its
properties to Voltaire’s equations, than Fredgolm’s equations at first kind.
To find solution of second initial - edge problem the formula of Theorem on boundary
gives resolute singular boundary integral equation. Let’s note, that the formula of theorem has
besides boundary meanings of function and its normal derivative also boundary meanings of
change speed of that function, all with time delay.
This distinguishes dynamical analog of Green’s formula from same formula for elliptical equations and complicates the usage of method of consequent approximations, which commonly is
used for boundary integral equation of elliptical edge equations. Besides integral region of under integral functions depends on time, that also essentially distinguishes those from boundary
integral equation for elliptical problems. This is new class of boundary integral equations in delay potentials, that needs special research with functional analysis methods.
LITERATURE
1. Vladimirov V.S. Equations of mathematical physics. - М.1981. (in Russian)
2. Polyanin А.D. Reference book on linier equations of mathematical physics. – М.: Fizmatlit,
2001. (in Russian)
3. Reference book on special functions. Ed. By М. Аbramovits, I. Stigan. - М: Nauka, 1979.
(in Russian)
4. Mikhlin S.G. Course of mathematical physics. – М.: Nauka, 1968. (in Russian)
5. Alexeeva L.A. Method of generalized functions in non steady edge problems for wave equation// Mathematical journal. V.6 (2006), №1 (19), pp.16-32. (in Russian)
6. Alexeeva L.A., Baegizova A.S. Generalized solutions of non steady edge problems for
Klein-Gordon-Fock equations // Evolution journal of open systems. №1. 2011. pp.12-21. (in
Russian)
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