We consider initial and boundary value problems for linear nonhomogeneous difference equations with constant coefficients. For such problems we compute the numerical values of the solutions using the discrete deconvolution. The method can be easily used in applications and implemented on computer.difference equations, initial and boundary value problems, discrete convolution and deconvolution

Mircea I CÎRNU

English

Research Papers in Economics

INITIAL AND BOUNDARY VALUE PROBLEMS 
FOR DIFFERENCE EQUATIONS 
 
Mircea I CÎRNU, PhD 
Dept. of Mathematics III, Faculty of Applied Sciences 
University "Politehnica" of Bucharest 
cirnumircea@yahoo.com 
 
Abstract:    We consider initial and boundary value problems  for linear nonhomogeneous difference 
equations with constant coefficients. For such problems we compute the numerical values of the solutions 
using the discrete deconvolution. The method can be easily used in applications and  implemented on 
computer.  
 
  Keywords:  difference equations, initial and boundary value problems, discrete convolution 
and deconvolution.  
  2000 Mathematics Subject Classification:  40-04, 40A05 
 
1. Discrete convolution and deconvolution  
 
Being given two finite sequences of numbers, of the same length,  ( ) k a a a a ; ; ; 1 0  =   and 
( ) k b b b b ;...; ; 1 0 = , one consider the (Cauchy) discrete convolution (see [2])   ( ) k c c c b a c ; ; ; 1 0  = ∗ = , 
of components  0 0 0 b a c = ,  ∑ = − = + =
k
j j j k k b a c b a b a c
0 1 0 0 1 1 , , .It is commutative, associative, 
distributive with respect to the addition of the sequences and has the unit  ( ) 0 ;...; 0 ; 0 ; 1 = δ .  
If a and c are known  and  0 0 ≠ a , we can determine b , such that    b a c ∗ = .  This sequence is 
called the deconvolution of  c by a, is denoted  a c b =  and has the components  
                    ( ) ( ) ∑
−
= − − = − = =
1
0
0
0 1 1
0
1
0
0
0
1
, ,
1
,
k
j j j k k k b a c
a
b b a c
a
b
a
c
b         
hence it can be calculated by the algorithm 
                   k c c c  1 0           k a a a  1 0  
                   0 0 1 0 b a b a c k     
   
k b
a
b a c
b
a
c
b 
0
0 1 1
1
0
0
0
−
= = /  0 1 1 b a c − 0 b a c k k −  
1 0 1 1 b a b a c k  −    
Particularly, one denote  a a δ =
−1 , hence 
1 − ∗ = a c a c . If   a  is a finite sequence and  b   a 
finite or infinite sequence, we denote by ( ) b a;   the sequence obtained by adjoining the two sequences`. 
The notions of convolution and deconvolution can also be considered, with the same definitions 
and notations, in the case of the infinite sequences.  
The convolution and the deconvolution operations considered above for finite sequences (of the 
same length) represent a truncate version of the usual operations for finite sequences, considered for 
example in MATLAB, by the instructions "conv" and "deconv" and which represent the multiplication 
and the division of the polynomials.  
 
2. The main result.    
 
We consider the nonhomogeneous linear difference equation       
  ,... 1 , 0 ,
0 = = ∑ = + − k b u a
n
j k k j j n   
 
with the  coefficients  , 0 0 ≠ a   n a a ,..., 1  and the right hand  terms  ,... ,..., , 1 0 k b b b . 
  We denote  ( )   ; 0 ; 0 ; ; ; ; 1 0 n a a a a =  and  ( )   ; ; ; ; 1 0 k b b b b = .     
 
  The unique solution   ( )   ; ; ; ; 1 0 k u u u u =   of the  equation  with known initial values  
1 1 0 , , , − n u u u  , is given by the deconvolution formula (see [1])       
  ( ) ( )= + = ∑
−
= − −    ; 0 ; 0 ; ; ; ; ; ; ; ; 0 1 0
1
0 1 1 0 0 1 0 0 n
n
j j j n a a b b u a u a u a u a u      
 
  ( ) ( ) ( ) = ∗ ∗ =
−
− −
1
1 1 0 1 1 0 ; ; ; ; ; ; ; a b u u u a a a n n          
 
     
+ ∗ + ∗ =
−
−
− 1
1 1 0 0
1 ) ; 0 ; 0 ; ; ; ; ( ) ; 0 ; ; 0 ( a a a a u a b n
n
      
 
  1
0
1
1
1
2 0 1 ) ; 0 ; 0 ; ; 0 ; ; 0 ( ) ; 0 ; 0 ; ; ; ; 0 (
−
−
−
−
− ∗ + + ∗ + a a u a a a u
n
n n        .     
 
Remark.  In the last form of  u  the sequence  ) ; 0 ; ; 0 (
1 − ∗a b
n
    is a particular solution of the 
nonhomogeneous difference equation is the sum between its particular solution   ) ; 0 ; ; 0 (
1 − ∗a b
n
     while 
the sum of the other terms represents the general solution, with arbitrary coefficient  1 0 , , − n u u  , of the 
homogeneous associated  equation        
 ∑ = + − =
n
j k j j n u a
0 0 ,    , 1 , 0 = k   
 
2.2. Initial value problem    
If we know the first n  components   1 1 0 , , , − n u u u  , called initial values, of the solution u  of the 
equation, the deconvolution formula gives the possibility to calculate as many components of u   as 
required. 
 
    Example 1.  The initial value problem formed by the linear difference equation   
,... 2 , 1 , 0 , 3 2 1 2 = = − − + + k k u u u k k k and the initial  conditions  1 1 0 = = u u , has  2 = n   , 
( )  ; 0 ; 0 ; 3 ; 2 ; 1 − − = a , ( )  ; 2 ; 1 ; 0 = b , 
 
( ) ( ) ( ) ( ) ( ) ( )= ∗ − = ∗ = b b u u a a c ; 1 ; 1 2 ; 1 ; ; ; 1 0 1 0 ( )  ; 3 ; 2 ; 1 ; 0 ; 1 ; 1 − .     
 
The deconvolution algorithm   
gives the sequence   ( ) ... ; 45 ; 14 ; 5 ; 1 ; 1 = = a c u   as numerical solution of the initial value problem.  
3. Boundary value problem 
 
  We now consider the problem of finding the solution   ( )   ; ; ; ; 1 0 k u u u u =  of the difference 
equation for which are known n components  
n k k k u u u , , ,
2 1  , called  boundary values. By equalizing 
these values with the corresponding components of the solution of the equation, given by the last form of 
the deconvolution formula, we obtain a linear algebraic system of order n, from which it is eventually 
possible to calculate the initial values  . , , , 1 1 0 − n u u u    Replacing these values in the deconvolution 
formula, we obtain the solution of the considered boundary value problem.     
  Example 2. We consider the same equation as in example 1, but now with the boundary 
conditions  409 , 45 6 4 = = u u . Because          
  ( ) = − − =
− − 1 1 ; 0 ; 0 ; 3 ; 2 ; 1  a ( ) ( )= − −   ; 0 ; 0 ; 3 ; 2 ; 1 ; 0 ; 0 ; 1       =
( )  ; 1640 ; 547 ; 182 ; 61 ; 20 ; 7 ; 2 ; 1 ,   
(here and in the following examples, the deconvolution algorithms is not effectively presented), we have     
 
( ) ( ) + ∗ + ∗ =
− − 1
1 0 0
1 ; 0 ; 0 ; ; ; 0 ; 0 a a a u a b u  ( ) = ∗
−1
0 1 ; 0 ; 0 ; ; 0 a a u   
 
( ) ( )
1
0
1 ; 0 ; 0 ; 2 ; 1 ; 0 ; 0
− − ∗ − + ∗ = a u a b  ( ) = ∗ +
−1
1 ; 0 ; 0 ; 1 ; 0 a u      
 
( )+ =  ; 135 ; 44 ; 14 ; 4 ; 1 ; 0 ; 0 ; 0 ( )+  ; 546 ; 183 ; 60 ; 21 ; 6 ; 3 ; 0 ; 1 0 u  
( )  ; 547 ; 182 ; 61 ; 20 ; 7 ; 2 ; 1 ; 0 1 u + .  
 
The boundary value conditions    
45 20 21 4 1 0 4 = + + = u u u ,   
409 182 183 44 1 0 6 = + + = u u u     
 
give  1 1 0 = = u u . Replacing these values in the above formula for u  , we obtain the same value for u  as 
in example 1,  namely   
( )  ; 1228 ; 409 ; 135 ; 45 ; 14 ; 5 ; 1 ; 1 = u  .  
 
4. Qualitative aspects of the boundary value problem 
  As will be seen in the example 3 below, a boundary value problem can  have a unique solution, 
an infinity of solutions or it can  have no solution.       
  Example 3. The difference equation   k k k b u u = + +2 ,   , 2 , 1 , 0 = k , with  ( )= =   ; ; ; ; 1 0 k b b b b
( )  ; 2 ; 0 ; 2 ; 0 ; 2 ; 0 ; 2 ; 0 ; 2 − −  , has   ( )  ; 0 ; 0 ; 1 ; 0 ; 1 = a , and  ( )  ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1
1 − − =
− a  , hence  ( ) ( ) ( ) = ∗ + ∗ + ∗ =
− − − 1
1
1
0
1 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 a u a u a b u  
( ) ( ) ( )    ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 6 ; 0 ; 4 ; 0 ; 2 ; 0 ; 0 1 0 − − + − − + − = u u  
a) If  3 , 0 4 3 − = = u u , one obtains  0 3 1 = − = u u  and  3 4 0 4 − = + − = u u  , hence  1 0 = u . The boundary 
value problem has the unique solution  ( )  ; 0 ; 5 ; 0 ; 3 ; 0 ; 1 ; 0 ; 1 − = u  .  
b) If  3 , 1 4 2 − = = u u , from  1 2 0 2 = − = u u  and  4 u 3 4 0 − = + − = u ,one obtains only  1 0 = u , 
hence the boundary value problem has an infinity of solutions given by the relation         
( ) ( )   ; 0 ; 1 ; 1 ; 0 ; 1 ; 0 ; 1 ; 0 ; 0 ; 5 ; 0 ; 3 ; 0 ; 1 ; 0 ; 1 1 − − + − = u u  , whe  1 u  is an arbitrary  parameter. 
       
c) If  3 , 1 4 2 − = ≠ u u   , one obtains both   1 0 ≠ u   and   1 0 = u , contradiction, hence the 
considered boundary value problem has no solution.     
References 
1.    M.I.Cîrnu, Solving Difference and Differential Equations by Discrete Deconvolution , UPB Sci. 
Bull., Series A,  Vol. 69, No. 1, 13-26, 2007     
 2. M.A.Cuenod, Introduction `a l’analyse impulsionelle. Principes et applications, Dunod, Paris, 1970.     

Introduction `a l’analyse impulsionelle. Principes et applications,

Solving Difference and Differential Equations by Discrete Deconvolution ,

INITIAL AND BOUNDARY VALUE PROBLEMS FOR DIFFERENCE EQUATIONS

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INITIAL AND BOUNDARY VALUE PROBLEMS FOR DIFFERENCE EQUATIONS

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