The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.


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 x a x x x x t t t = + + ∫ y F y g K y (1.1) where the kernel T  T  T  T  1  2 , , , 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.

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 we could construct higher order polynomial spline functions with higher degree of smoothness. More specifically, ( ) are called one dimensional B-splines, which are polynomial splines and have   small supports  1  1  ,  2 2 , 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 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]).
exists and is bounded, let h be a real number, let , , , 3 .

A Univariate C 2 Cardinal Spline
By cardinal conditions (cf. [7]), we mean, let ( ) L x be a function, The cardinal spline that was originally given in [9]  , of its support. Furthermore, from direct calculation we deduct the following two propositions (cf. [9]).

( )
3 L x be the cardinal spline constructed above, then where , , α β γ are any complex numbers.
, f x C ∈ −∞ ∞ and be bounded, let h be a real number, let

Numerical Methods Solving Systems of Integral Equations
Method 1-V for solving the system of nonlinear Volterra integral equations which is a simple system of ( ) . Notice that this is a nearly triangular system and it is solvable (the solution may not be unique because it is not linear): , 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 is the exact solution of Equation (1.1).
Example 2 Given the system of integral equations

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 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.