The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions ()
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. There are also some problems that can be expressed only in terms of integral equations. Abundant papers have appeared on solving integral equations, for example, Polyanin summarized different solutions of integral equations in [1] and [2] [3] published in 2013 and 2016. In [4] [5] and [6], we discussed numerical methods using cardinal splines in solving systems of linear integral equations. In this paper we are going to explore the applications of cardinal splines in solving nonlinear systems of integral equations.
We are interested in the systems of Volterra integral equations of the second kind
(1.1)
where the kernel
and
are known functions, and
is to be determined;
are known functions in
.
This paper is divided into six sections. In Section 2 and 3, two univariate cardinal continuous splines on small compact supports are presented. In Section 4, the applications of cardinal splines on solving integral equations are explored. The unknown functions are expressed as linear combinations of horizontal translations of a cardinal spline function. Then a system of equations on the coefficients is deducted. We can solve the system and a good approximation of the original solution is obtained. The sufficient condition for the existence of the inverse matrix is discussed and the convergence is investigated. In Section 5, the numerical examples are given. The non-linear system on unknowns is solved and an accurate approximation of the original solution is obtained in each case. Section 6 contains the concluding remarks.
2. Cardinal Splines with Small Compact Supports
Since the paper [7] by Schoenberg published in 1946, spline functions have been studied by many scholars. Spline functions have excellent properties and applications are endless (for example, cf. [8]). The spline functions on uniform partitions are simple to construct and easy to apply, and are sufficient for a variety of applications.
The starting point is frequently the zero degree polynomial B-spline, with the integral iteration formula
(2.1)
(2.2)
we could construct higher order polynomial spline functions with higher degree of smoothness. More specifically,
has the expression
(2.3)
are called one dimensional B-splines, which are polynomial splines and have small supports
, i.e.
for
or
, and excellent traits (cf. [8]). In my previous papers [4] and [5], low degree orthonormal
spline and cardinal splines functions with small compact supports were applied in solving the second kind of Volterra integral equations. In this paper we use the notation
.
Let
. It is proved that
Notice that this particular B-spline is also a cardinal spline, therefore it is straightforward to apply it in interpolations. As far as the convergence rate of interpolation is concerned, we have the following proposition (cf. [9] [10] and [11]).
Proposition 1. Given that
,
exists and is bounded in
. Let n be an integer,
, let
,
,
then
If
and
exists and is bounded, let h be a real number, let
,
then
(2.4)
3. A Univariate C2 Cardinal Spline
By cardinal conditions (cf. [7]), we mean, let
be a function,
be interpolation points, then
(3.1)
The cardinal spline that was originally given in [9] is based on
from (1.1) using the similar process as in Section 2. Let
(3.2)
Then
satisfies the above cardinal condition when
. Notice that by the construction,
.
for
.
is a polynomial of degree ≤ 5 in each subinterval
of its support. Furthermore, from direct calculation we deduct the following two propositions (cf. [9]).
Proposition 2. Let
be the cardinal spline constructed above, then
where
are any complex numbers.
Proposition 3. If
, let
, n be an integer, let
,
then
If
and be bounded, let h be a real number, let
,
then
4. Numerical Methods Solving Systems of Integral Equations
Method 1-V for solving the system of nonlinear Volterra integral equations
As for the Volterra integral Equations (1.1) we solve it in an interval
. Again we let
,
,
. Furthermore, by pluging in
,
,
, we get
Let
, we arrive at
(4.1)
(4.2)
which is a simple system of
nonlinear equations of unknowns
. Notice that this is a nearly triangular system and it is solvable (the solution may not be unique because it is not linear):
where
.
Proposition 4. Given the Equation (1) and that
,
and
exists and is bounded in
,
,
. Furthermore,
and
satisfies the condition:
where
. Let n be an integer,
, let
,
,
satisfy the nonlinear system (2.2)
then
where
is the exact solution of Equation (1.1).
Method 2-V for solving the Volterra integral equation
To improve the approximation rate, we apply the spline function
. Again we let
,
. Furthermore, let
be the cardinal spline given in Section 3, and
, where
where
. Let
be unknown coefficients to be determined and
,
. Plug into the integral Equation (1.1), then we have
Let
, we arrive at (
)
(4.3)
which is still a relatively simple system of equations. For the convergency rate of solution of the Volterra integral Equations (1.1), we have a similar result.
Proposition 5. Given that
,
and
exists and is bounded in
,
,
. Furthermore,
satisfies the condition:
where
. Let n be an integer,
, let
,
;
satisfy the system (4.2)
then
where
is the exact solution of Equation (1.1).
5. Numerical Examples
Example 1. Given the system of integral equations
where
,
, and
are unknown functions.
Let
,
,
,
,
,
,
we get
where
,
. Let
, we arrive at
Solve the above nonlinear system and we obtain
(In the paper [12], the error is 0.02579 for
)
Method 2-V, let
,
. The original system becomes
Applying L3, we still Let
,
,
,
,
,
,
, where
, (we use the given integral equation to find that
,
,
,
). Similarly,
. In addition, we let
and
. Plug into the integral equations, we obtain the nonlinear system:
the solution of the system is
Furthermore, we solve
,
,
. The results are
The error < 2 × 10−5. Our method is much better.
Example 2 Given the system of integral equations
where
,
. Let
,
,
.
,
,
,
,
, where
are unknown functions. we get
where
,
. Let
, we arrive at
Solve the nonlinear system and we obtain:
6. Conclusion
The proposed method is a simple and effective procedure for solving nonlinear Volterra integral equations of the second kind. The methods can be adapted easily to the Volterra integral equations of the first kind, which have the form
, where the upper limit of the integration is a variable.
The methods can also be extended to the Fredholm and Volterra integral equations of the first kind or the second kind, where the integral is on an infinite set. The higher degree cardinal splines could also be applied to non-linear integral equations; the resulting system of coefficients will be a little more complicated non-linear systems, which takes more time and effort to solve. Compared with the recent paper [2], our method is more effective.
Acknowledgements
The research is partially supported by the research funds of the University of La Verne, and by key natural science foundation of Anhui education commission (NO.KJ2017A568), natural science foundation of Bengbu University (NO.2018CXY045; 00009134).