SOME PROBLEMS IN THE THEORY
OF SINGULAR PERTURBED
INTEGRO-DIFFERENTIAL EQUATIONS
Ya. Goltser
Department of Mathematics and Computer Sciences, Ariel University Center of
Samaria, Israel
Abstract. We examine a finite-dimensional integro-differenzial system of Volterra
equations continuously dependent on a small vectorial or scalar parameter µ
t
dx
(1) A(  )  B(  ) x  f (t ,  )   K (t , s,  ) g ( s, x( s),  )ds,
dt
0
with the kernel of the following form

(2) K (t , s,  )   C j ( ) F j (t ,  )G j (s,  ),
j 1
in case when A(0) singular matrix with ind A(0)=k>0.
We will discuss problems related to applications of the reduction method for
investigation and solution of systems (1), (2).
A constructive approach to analysis and solving of linear singular perturbed system
ODE is proposed. This technique is used the Drazin inverse of singular matrices and
provides a way of estimating and establishes a link between singular and singular
perturbed systems, structural determinations of a boundary layer into solution of
singular perturbed system.
1. INTRODUCTION
We consider the following system of Volterra type IDE insoluble with respect to the
derivative
t
dx
(1.1) A(  )  B(  ) x  f (t ,  )   K (t , s,  ) g ( s, x( s),  )ds,
dt
0
which is continuously dependent on parameter μ. It is assumed that the matrix A(0) is
singular, and its index k=ind A(0)≥1.
Such systems have been studied from various standpoints in the case of
(1.2) A( )  diag ( Er , El ) ,
E is an identity matrix, μ is a small scalar parameter (1-5).Matrices of the form (1.2)
are singular at µ=0 and their index k=1.
Because of this, the asymptotic of the solution of the initial-value Cauchy problem
contains boundary functions Πx(τ) satisfying the estimates of the following form:
t
(1.3) x( )  c exp(   ), c  0,   0,  

See (1),Chapter 3.
In the present paper we suggest an outline of the analysis of certain classes of
systems of the form (1.1) based on the combination of the methods of IDEs reduction
to ODEs and integration of singular systems of ODEs. First we describe in Section 2
the method of IDE system reduction to ODE system (finite or countable one). In
Section 3 the method of integrating finite-dimensional singular linear systems using
inverse Drazin's matrices is developed.
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It will be evident from the results of Section 3 that in case where μ is a vectorial
parameter and the index of A(0) matrix k≥ 1, the structure of boundary functions in
the asymptotic of the solutions should be different from (1.3).
2. REDUCTION OF IDES TO A COUNTABLE SYSTEM OF ODES
In this section we present the principal data related to the application of reduction
method to singular IDE systems. Here we use the notions of the reduction method set
forth earlier in (6, 7, 8)for regular IDE systems.
We assume that the kernel K(t,s,μ) in (1.1) can be represented in the form

(2.1) K (t , s,  )   C j ( ) F j (t ,  )G j (t ,  ).
j 1
Following (9), we present the necessary information from the theory of linear
differential equations. Let the system of equations
dx
(2.2)
 P(t ) x  f (t ), x(t 0 )  x0
dt
be specified. The solution of Cauchy problem for (2.2) can be written in the form
t
(2.3) x(t )   tt0 ( P) x(t 0 )   K (t , s) f ( s)ds, K (t , s)   tt0 ( P)[ ts0 ( P)] 1 ,
t0
is a matrizant of a homogeneous system.  tt0( P ) where
The matrizant can be represented in the form of a multiplicative Volterra integral:
t
(2.4)  tt0 ( P )   ( E  P (t )dt ) .
t0
Let us examine a multiplicative derivative
dX (t ) 1
X (t ).
dt
Both operations are mutually inverse, i.e. if
(2.6) Dt X (t )  P(t )
then
(2.5)
(2.7)
Dt X (t ) 
t
X (t )   ( E  P(t )dt )  C , C  X (t 0 ).
t0
Consequently, if Cauchy's presentation (2.3) is specified, then we can see from (2.6)
and (2.7) that the system (2.2) having the solution (2.3) is determined by the matrix
P(t). Here P(t) is unambiguously defined using (2.6) as a multiplicative derivative of
the matrizant
(2.8) X (t )   tt ( P).
0
Let us revert to IDE system (1.1) with the kernel (2.1). We write
(2.9)
K j (t , s,  )  F j (t ,  ) F j1 (( s,  ) K j ( s, s,  ), K j ( s, s,  )  F j ( s,  )G j ( s,  )
and set
(2.10)
Pj (t ,  )  Dt F j (t ,  ) 
dF j (t ,  )
dt
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F j1 (t ,  ).
Then for nonhomogeneous system
dy j
(2.11)
 Pj (t ,  ) y j  f j (t. )
dt
we obtain the following solution of the Cauchy problem y(0,μ)=0
t
(2.12)
y j (t ,  )   F j (t ,  ) F j1 ( s,  ) f j ( s,  )ds.
0
Assuming that in (2.11) and (2.12)
(2.13) f j ( s, x( s),  )  K j ( s, s,  ) f ( x( s),  ),
and introducing new variables
t
(2.14)
y j (t ,  )   K j (t , s,  ) f j ( x( s),  )ds, y j (0,  )  0.
0
We obtain the following countable ODE system instead of the IDE system (1.1):

dx
A(  )
 B(  ) x  G (t ,  )   C j (t ,  ) y j
dt
j 1
(2.15)
dy j
 Pj (t ,  ) y j  f j (t , x,  ), j  1,2,...
dt
Thus, if the kernel in (1.1) has the presentation (2.1),then the system of IDE is
reduced to a countable system of ODE.In particular, expansions of the form (2.1)
have Gilbert-Schmidt's kernels.
Let us single out certain particular cases:
a) Autonomous case. Let us find the condition of the independence of matrices in
(2.15) on t. PJ (t ,  )
It follows from the equality determining P that
dF
(2.16) Pj F j  J
dt
and then
F j (t ,  )  exp( Pj (  )t )
Thus, as would be expected, the kernel structure in the case of autonomous linear part
in (2.15) has the following form
(2.17) K j (t , s,  )  exp[ Pj (  )(t  s)].
b) Periodic case. Let us clarify the kernel structure in the assumption that is an ωperiodic matrices. Pj (t ,  )
From the equation (2.16) using Floquet's theory, we come to the conclusion that
in this case the kernel contains the matrix
(2.18) F j (t ,  )   j (t ,  ) exp( (  )t ),  j (t   )   j (t ).
Thus, the kernel has the form
(2.19) K j (t, s,  )   j (t ,  ) exp[ ( )(t  s)] j 1 (s,  ) K j (s, s,  )
and
dF j t ,  1
 d j (t ,  )

(2.20) Pj (t ,  ) 
F j (t ,   
  j (t ,  ) j (t ,  )  j1 (t ,  ).
dt
dt


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The special character of the problem under study consists in the fact that the initial
system of IDEs and the corresponding system of ODEs represent singular systems,
since detA(0)=0. Therefore, when using the proposed reduction method, it becomes
necessary to investigate two aspects:
1) To study the characteristics of the solution with respect to t.
2) To study the characteristics of the solution x(t,µ) with respect to µ.
Here it is important to construct the asymptotic of the solution with respect to μ. This
issue is discussed in Section 3 on the example of linear systems.
3. INTEGRATION OF SINGULAR LINEAR SYSTEMS
3.1 Drazin's inverse matrix
We remind some notions connected with the inversion of singular square matrices
(4, 13), which will be necessary hereinafter. Let an A-n×n-matrix be specified.
Definition 3.1 A matrix A D is called an inverse Drazin's matrix of A if
(3.1)
AX  XA, XAX  X , XA k 1  A k , rank A k 1  rank A k , (k  ind A).
Note, k is the dimension of the greatest nilpotent block in the Jordan presentation of
the matrix A. The inverse Drazin's matrix is unique and always exists. If k=0 then
A D  A 1 .
3.2. Principal lemma.
We consider a singularly perturbed linear system of ODE
dx
(3.2) A(  )
 B(  ) x  f (t ,  ), 0     , det A(0)  0, det B(0)  0
dt
Let us pass from the system (3.2) to a system solvable with respect to the variable x:
dx
(3.3) x  M (  )   (t ,  ), M (  )  B 1 (  ) A(  ),  (t ,  )   B 1 (  ) f (t ,  ).
dt
We multiply (3.3) by ( E  M 0D ), ( M 0  M (0)) . Then, applying equation (3.3) k times
(k=indM(0)), we obtain an equality


dkx
(3.4) ( E  M 0D M (  )) x  ( E  M 0D M (  )) M k (  ) k  T k 1 ( (t ,  )) ,
dt


where T is an operator:
T 0   , T  M ( )    , T l  T (T l 1 ).
We put
(3.5) Z (  )  H (  ) M k (  ), H (  )  E  M 0D M (  ).
The principal property of the matrix Z(μ) is
(3.6) Z (0)  ( E  M 0D M 0 ) M 0k  0,
i.e. all the elements of Z(µ) matrix are infinitesimal functions in the neighborhood of
the point µ=0.
Lemma 3.1. Any system of the form (3.2), where A(0)is a singular n×n matrix with
the index k≥1 and a reversible matrix B(μ) is reducible to a system of equations of the
k-th order of the form
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dkx
 H (  ) x  H (  )T k 1 ( (t ,  )),
k
dt
where Z (0)  0, H (  )  E  M 0D M (  ), M 0  B 1 (0) A(0).
(3.7) Z (  )
3.3 Integration of a linear system in a regularly singular case
Lemma 3.1 shows that in order to integrate an n-dimensional system (3.2) with a
singular matrix A(µ), we can always pass to a problem of integrating a system of
linear equations of the k-th order with an infinitesimal matrix coefficient of the
derivative. Here we can come across different cases with different character of the
asymptotic of solutions with respect to µ. Let us single out and examine the case that
we call a regularly singular one. It can be characterized as follows:
Condition α. There exists an infinitesimal positive continuous function α(µ),such
that the matrix Z(μ) possesses the property lim  ( )  0,
 0
1
Z (  )  Z 0 det Z 0  0.
 0  (  )
As clearly follows from further reasoning, α(μ) plays the principal part in the
asymptotics of solutions. We introduce a substitution
(3.8) lim
(3.9) t   (  )k  .
Then, taking into account (3.8), instead of a nonhomogeneous system (3.7), we obtain
a system of the form
dkx
(3.10)
 C (  ) x   (t ,  ), C (  )  ( Z 0  (  )) 1 H (  ),
k
dt
where the matrix C(μ) has zero eigenvalues, and C(μ)-small eigenvalues.
Let us examine the structure of the solutions of a homogeneous system
dkx
(3.11)
 C (  ) x.
dt k
We limit ourselves by the case where the matrix C(μ) has n different eigenvalues
(3.12) 1 (  ),  2 (  ),...,  n (  ),  j (  )   k (  ),   0
Theorem 4.2. Let the system (4.18) be such that the matrix C(μ) has eigenvalue
(3.12) with the corresponding eigenvectors
l (1) ( ), l ( 2) ( ),..., l ( n ) ( ).
Then the general solution of the system (3.11) has the form
1
n
(3.13)
x( ,  )   l
s 1
1
k
k
(s)
(  ) C sj exp(  sj (  )) ,  sj (  )  [  s (  )] , j  1,2,..., k ,
j 1
C sj - arbitrary constants.
Proof. Evidently, (3.13) represents a linear combination of partial solutions.
Assuming that
(3.14) x( ,  )  l ( s ) ( ) exp  sj { )
and substituting x(τ,µ) into (3.11), we obtain
(3.15) l ( s ) ( )  sjk { )  C( )l ( s ) ( ),
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(  sjk ( )   s ( ), s, j  1,2,..., n) or, otherwise
(3.16) (C (  )   5 (  ) E )l ( s ) (  )  0
It is clear from (3.16) that (3.14) is the solution of the system (3.15).
Let us demonstrate that for any initial conditions
(3.17)   0, x(0)  x0 , x (0)  x0 ,..., x0( k 1) (0)  x0( k 1)
the Cauchy problem has a solution (3.13) at a certain choice of arbitrary constants .
Assuming τ=0 in (3.13), we obtain
n
k
s 1
j 1
(3.18) x0   l ( s ) (  ) C sj
Proceeding from (3.18) and expanding the vector x (0) in the orthogonal basis so that
n
x0   l ( s ) C s0
s 1
we obtain
k
(3.19)
C
j 1
sj
 C s0 , s  1,2,..., n,
0
s
where C are known numbers.
Differentiating the equality (3.13) p times and assuming τ=0 we obtain for any
p=1,2,...,k-1 following:
n
k
s 1
j 1
(3.20) x0( p )   l ( s ) (  ) C sj  sjp (  )
and, as above, we can find from (3.20)
k
(3.21)
C
j 1
sj
 sjp ( )  C s( p ) ,
where C s( p ) are known numbers, coordinates of vector (3.20) in the basis of
eigenvectors.
The system of equations (3.21), where p=0, 1,..., k-1, is decomposed into n
subsystems with the dimension k :
k
(3.22)

j 1
p
sj
C sj  C s( p ) , p  0,1,..., k  1
with s being fixed, s=1, 2,..., n.
The determinant of the system (3.22) is nonzero at any s (Vandermonde determinant).
Thus, (3.22) has a unique solution which determines the solution of the Cauchy
problem. Thus, (3.13\ is a general solution of the system (4.18).□
Reverting to the initial n-dimensional system, we can readily make sure of the
following:
Corollary 3.1 Let the initial conditions (3.17) satisfy conditions
(3.23) H (  ) x0( p 1)  H (  ) M (  ) x0( p ) , p  1, 2, ..., k  1, H (  )  E  M 0D M (  ).
Then the solution (3.13) corresponding to these conditions represents the solution of
the initial system at μ≠0.
It follows from (3.13) that the boundary layer structure in the solution x(t,μ) is
described by functions of the type
 sj (  )
(3.24) exp
t
k  ( )
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where  sj (0)  0.
3.4 Singular systems integration at µ=0.(4,13)
Let us examine the system (3.2) at μ=0 (substituting y instead of x)
dy
(3.25) A0
 B0 y  f 0 (t ), k  ind A 0 , B01 .
dt
We pass to a system
dy
(3.26) y  M 0
  0 (t ), M 0  B01 A0 ,  0   B01 f 0 .
dt
: Multiplying (3.39) by H 0  E  M 0D M 0 and applying (3.26) k times, we obtain


(3.27) H 0 y  H 0 M 0k y ( k )  T k 1 ( 0 ) .
Since M  M M
k
0
D
0
k 1
0
(3.27) has a zero matrix and the following equality is valid:
(3.28) H 0 y  H 0T ( 0 )
We obtain from (3.28)
(3.29) y  M 0D M 0  H 0T k 1 ( 0 ).
k 1
Let us find M 0D M 0 y . We can write:
dy
(3.30) M 0
 y  0 ,
dt
or, multiplying by M 0D and keeping in mind the equality
M 0D  M 0D M 0 M 0D  M 0D ( M 0D M 0 )
we obtain
d
(3.31)
( M 0D M 0 y )  M 0D ( M 0D M 0 y )  M 0D 0 (t )
dt
It follows from (3.31) that
t
(3.32) M 0D M 0 y(t )  exp[ M 0D t ]( M 0D M 0 y0   exp[ M 0D (t  s)]M 0D 0 ( s)ds
0
From (3.29) and (3.32) we obtain
t
(3.33)
y (t )  exp[ M 0D t ]M 0D M 0 y 0   exp[ M 0D (t  s )]M 0D 0 ( s )ds  H 0T k 1 ( 0 )
0
Clearly, the system (3.25) is not always soluble. We obtain from (3.33) at t=0 a
requirement for admissible initial conditions ensuring the solubility of the problem
Cauchy:
(3.34) y 0  M 0 M 0D y 0  H 0T k 1 ( 0 (0)
3.5 Example. The case of k=1.
Let indA(0)=1 in a homogeneous system (3.2). Then the system (3.2) is reducible at
μ≠0 to the form
dx
(3.35) Z (  )
 H (  ) x.
dt
Introducing variables t=α(μ)τ we obtain:
dx
(3.36)
 C (  ) x, C (  )  ( Z 0  (  )) 1 H (  ).
dt
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At µ=0 the system
dx
(3.37) A0
 B0 x, A0  A(0), B0  B(0)
dt
has a solution
(3.38) y (t )  exp[ M 0D t ] y 0 , H 0 y 0  0.
We are seeking a relationship between the solutions of the systems (3.35) and
(3.37).We put
(3.39) x(t ,  )  y (t )  z (t ,  ), y (t )  exp[ M 0D t ]M 0 M 0D y 0
Then after the substitution t=α(µ)τ, we obtain the following equation for z(t,µ):
dz
(3.40)
 C (  ) z   ( ,  ), lim  ( ,  )  0.
 0
d
Let  s (  ), s  1,..., n be the eigenvalues of the matrix C(μ).
It can be easily seen that a sufficient condition for equality
(3.41) lim x(t ,  )  y(t )
 0
is requirement
Re  s (  )  0, s  1,..., n,   0
in the neighborhood of the point µ=0.
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