Solving Systems of Volterra Integral Equations with Cardinal Splines ()
1. Introduction
Integral equations appear in many fields, including dynamic systems, mathematical applications in economics, communication theory, optimization and optimal control systems, biology and population growth, continuum and quantum mechanics, kinetic theory of gases, electricity and magnetism, potential theory, geophysics, etc. Many differential equations with boundary-value can be reformulated as integral equations. One example given in this paper is to use a system of integral equations to solve a third order differential equation. There are also some problems that can be expressed only in terms of integral equations. Scores of papers have appeared on solving integral equations, for examples, cf [2] - [4] .
2. Cardinal Splines with Small Compact Supports
Since the paper [5] by Schoenberg published in 1946, spline functions have been studied by many scholars extensively. Spline functions have excellent features and applications are endless (for examples, cf [6] ). The spline functions on uniform partitions are simple to construct and easy to employ, and are sufficient for a variety of applications.
The starting point is frequently the zero degree polynomial B-spline, an integral iteration formula could be used to construct higher order spline functions with higher degree of smoothness, i.e. let
![](//html.scirp.org/file/8-1720389x6.png)
(1)
has the global expression
are called one dimensional B- splines, which are polynomial splines with small compact supports
, (i.e.
for
), and excellent traits (cf [6] ). In my previous papers (cf [1] [7] - [10] ), low degree orthonormal
spline and cardinal spline functions with small compact supports were applied in solving the second kind of linear Fredholm and Volterra integral equations.
By cardinal conditions (cf [5] ), we mean, let
be an interpolation function,
be interpolation points, then
![](//html.scirp.org/file/8-1720389x16.png)
The lowest degree continuous cardinal spline is
. To achieve higher degree of approximation, we need the cardinal spline functions with higher degree of smoothness. We employ splines that were developed in my previous papers.
The cardinal spline
that was originally given in [11] is based on
from (1) using the similar integral process. Let
![](//html.scirp.org/file/8-1720389x20.png)
Then
satisfies the above cardinal condition when
Notice that by the
construction, ![]()
for
is a polynomial of degree
in each subinterval
of its support. Furthermore, we acquired nice approximation properties (cf [11] ).
To achieve higher degree of smoothness, we employ
(cf [6] ) and constructed (cf [11] )
![]()
It is a simple calculation to check the cardinal condition is satisfied. The
has the same support as
, which is
. However, each polynomial in the sub-intervals of
has higher degree than the
ones for
, Let
The better approximation properties hold for
(cf [11] ).
3. Numerical Methods Solving the System of Volterra Integral Equations
In this and next two sections, we are concentrating on the second kind system of linear Volterra integral equations
(I2)
where
, ![]()
3.1. Method 1-V for Solving the System of Volterra Integral Equations
As for the Volterra system (I2), we solve it in an interval
. Again we let
Furthermore, plug
(funs 1)
in (I2), we get
![]()
Let
, we arrive at for ![]()
(S4)
which is a simple system of
S linear equations on the unknowns
. For the convergency rate of solution of the Volterra system (I2), we have
Proposition 1. Given that
,
,
and
exist and are bounded in
,
![]()
and
exist and are bounded in
. Further
more
satisfies the condition:
![]()
where
. Let
satisfy the linear system (S1) and
; then
![]()
where
is the exact solution of the system of Equation (I1).
The proof is very similar to the proof of the following Proposition 3, so we skip it.
3.2. Method 2-V for Solving the Systems of Volterra Integral Equations
To achieve higher approximation rate, we plug
(funs 2)
into (I2), where extra function values still follow the conditions (cond1) and arrive at
![]()
Let
, By the similar reasons as above, we conclude that
(S5)
which is still a relatively simple system of linear equations.
Remark If the integral equation (I2) has a unique solution, then the linear system (S5) is consistent. Furthermore
![]()
(where
)
approximates the solution of the system (I2) with a rate of
, similar as in the Preposition 6.
Proposition 2. Given that
, ![]()
and
exist and are bounded in
, ![]()
exist and are bounded in
. Furthermore,
satisfies the condition:
![]()
where
. Let
satisfies the linear system (S2) and
,
. Let
then
![]()
where
are the exact solution of the Volterra system (I1).
The proof is very similar to the proof of the following Proposition 3, so we skip it.
3.3. Method 3-V for Solving the Systems of Volterra Integral Equations
To achieve higher approximation rate, we let
(funs 3)
again we plug into (I2), where extra function values still follow the conditions (cond2) cf [9] , and we arrive at
![]()
Let
, by the similar reasons as above, we conclude that,
(S6)
which is still a relatively simple system of linear equations.
Remark If the integral Equation (I2) has a unique solution, then the linear system (S6) is consistent. Furthermore
![]()
approximates the solution of the integral Equation (I2) with a rate of
as in the following Preposition. Where
![]()
![]()
![]()
Proposition 3. Given that
, ![]()
and
exist and are bounded in
, ![]()
exist and are bounded in
. Fur-
thermore,
satisfies the condition:
![]()
where
. Let
satisfies the linear system (S3) and
![]()
![]()
![]()
![]()
![]()
Let
then
![]()
where
are the exact solution of the Volterra system (I1).
Proof. Let
,
![]()
![]()
where the coefficients are the solutions of above linear system (S3), and
![]()
![]()
![]()
, then
(I1)
![]()
Plug in
![]()
Therefore
![]()
4. Numerical Examples
Example 1. Consider
,
![]()
Let ![]()
![]()
Apply Method V-1 and solve the linear system, we obtain:
![]()
To achieve higher degree of accuracy, we apply Method V-3 and obtain :
![]()
Compare with the exact solution:
![]()
the error is
for
.
Example 2. Given a system of linear integral equations: for
,
![]()
![]()
Let
We apply the Method V-1, where
![]()
are unknown coefficients to be determined. Solve the system of
linear equations on unknowns
obtain
![]()
and
![]()
Since the solution is
Comparing with the exact solution, the error is
for
.
Example 3. Consider the third order differential equations:
,
with the initial condition: ![]()
Let
then we could transfer the equation to a system of integral equations:
,
,
![]()
Let ![]()
![]()
![]()
![]()
We apply Method V-2 and solve the linear system and arrive at the solution:
![]()
Since the exact solution is
![]()
Compare with the exact solution, the error
.
5. Conclusions
The system of first kind of linear Volterra integral equations has the form
(I4)
where
![]()
They can easily be transformed to the system of second kind of linear Fredholm and Volterra integral equations (cf [3] ). We can apply the similar method to solve the first and second kind Fredholm integral systems. So the proposed methods are simple and effective procedures for solving both linear system of Fredholm and Volterra integral equations.
The orthonormal and cardinal splines could also be applied to non-linear integral equations; the resulting system of coefficients will be a non-linear system, which takes more time and effort to solve. The convergence rate could be higher if we apply more complicated orthonormal or cardinal spline functions.
Acknowledgements
The work was partially funded by the National Natural Science Foundation of China under Grant no. 1471093, the Doctoral Program Foundation of the Ministry of Education of China under Grant no. 20110111120026, the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15, the Key Project Foundation of Scientific Research, Education Department of Anhui Province under Grant no. KJ2014ZD30.
The authors thank the family members, the colleagues and administrators of the
University
of LaVerne for their encouragement and support on this research.