The modern Galerkin method for integral equations of the mixed type
Lechosław Hącia
Institute of Mathematics, Faculty of Electrical Engineering,
Poznań University of Technology
60 - 965 Poznań, ul. Piotrowo 3A, e-mail: [email protected]
We can see that integral equations of the mixed type
t
u ( x, t )  f ( x, t )    k ( x, t , y, s)u ( y, s)dyds .
(1)
0 M
or shortly
u = f + Ku
(2)
play very important role in epidemiology, mechanics, electromagnetics and technology. Presented equations
arise also in the heat conduction theory [5,7} and the mathematical modelling of the spatio-temporal
development of an epidemic [1,2]. The spread of the disease in the given population can be described by the
following mixed integral equations. Some initial-boundary problems for a number differential partial equations
in physics are reducible to the considered integral equations. In this paper projection method of the Galerkin type
is proposed which leads to a system of Volterra integral equations. Consider equation (1) in space-time, where f
is given function in domain D  M  [0, T ] (M – a compact subset of m-dimensional Euclidean space) and u is
unknown function in D. Given kernel k is defined in domain   {( x, t, y, s): x, y  M,0  s  t  T} . Numerical
methods for studied equations were presented in [1, 3-7]. Convergence and error’s estimate of the method will be
presented
The modern method of the Galerkin type
In this section we propose a projection method for equation (1) leading to solve a system of Volterra linear
integral equations. Approximate solution of (1) we seak in the form
u n  x, t  
 a t  x
-
j
(3)
j
j 1
for  x, t   D , D  M  0, T  , where:
-
n
  is an orthonormal and complete basis in w L M  ;
a  is a solution to system of the following Volterra integral equations
2
j
j
a j t   f j t  
n
t
  k t , s a s ds ,
jk
j  1, 2, ..., n
k
(4)
k 1 0
with
f j t  
 f x, t  x dx ,
j  1, 2, ..., n
j
M
k jk t , s  
  k x, t, y, s  x  y dydx ,
j
k , j  1, 2, ..., n .
k
MM
Lemma
If
f  L2 D  and k  L2  ,    x, t , y, s  : 0  s  t  T ; x, y  M  , then function (3) is a
unique solution in the space
L2 D  of the equation
t
u n x, t   f n x, t     k n x, t , y, s u n  y, s dyds ,
(5)
0M
with
f n  x, t  
n
 f t  x  ,
k
k 1
k
(6)
k n  x, t , y , s  
n
n
 k t, s  x   y  .
jk
j
(7)
k
j 1 k 1
Proof. Let us notice that
u n  x, t  
n
 u t  x  ,
k
(8)
k
k 1
where
u k t   f k t  
n
t
   k t , s   y u  y, s dyds ,
jk
j
k  1, 2, ..., n .
n
(9)
j 1 0 M
By orthonormality of  k  we obtain the Volterra system of integral equations
u k t   f k t  
n
t
  k t , s u s ds ,
jk
j
k  1, 2, ..., n .
(10)
uk 
in space L2 0, T  such
j 1 0
From assumptions and Volterra theory it follows, this system has unique solution
that
u k t   a k t  for every k  1, 2, ..., n .
Remark
Equations (5) we can rewrite in the operator form
un  f n  K nun ,
where
(11)
K n is Volterra-Fredholm integral operator of form (2) determined by the kernel k n (7).
Theorem
If f  L2 D  and k  L2  , then sequence u n  defined by formula (3) converges in the space
unique solution of equation (1) and the estimate error
c
un  u L2 D  
1  c
holds with
c  I  K 
1
Proof is based on the equality
ff
n
n L2  D 
and
 u L2 D  n
L2 D  to

 n  kn  k
L2  
.
un  u  f n  f  K n  K un  K un  u .
References
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