Solving Systems of Volterra Integral Equations with Cardinal Splines

This work is a continuation of the earlier article [1]. We establish new numerical methods for solving systems of Volterra integral equations with cardinal splines. The unknown functions are expressed as a linear combination of horizontal translations of certain cardinal spline functions with small compact supports. Then a simple system of equations on the coefficients is acquired for the system of integral equations. It is relatively straight forward to solve the system of unknowns and an approximation of the original solution with high accuracy is achieved. Several cardinal splines are applied in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined and the convergence rate is investigated. We demonstrated the value of the methods using several 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.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].

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 ( ) ( ) ( ) ), 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 ( ) L x be an interpolation function, { }, 0, 1, 2, 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.

3
L x that was originally given in [11] is based on ( )

3
B x from (1) using the similar integral process.Let satisfies the above cardinal condition when , 0, 1, .
The better approximation properties hold for ( )

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 where

Method 1-V for Solving the System of Volterra Integral Equations
As for the Volterra system (I2), we solve it in an interval ( ) , .
which is a simple system of ( ) ( ) satisfy the linear system (S1) and 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.

Method 2-V for Solving the Systems of Volterra Integral Equations
To achieve higher approximation rate, we plug , .
, where extra function values still follow the conditions (cond1) and arrive at .
= , By the similar reasons as above, we conclude that , 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 , ,  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.

Method 3-V for Solving the Systems of Volterra Integral Equations
To achieve higher approximation rate, we let , .
again we plug into (I2), where extra function values still follow the conditions (cond2) cf [9], and we arrive at .
= , by the similar reasons as above, we conclude that, , 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 ( )

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O h as in the following Preposition.Where ( )  ,  are the exact solution of the Volterra system (I1).Proof.Let ( ) , , , where the coefficients are the solutions of above linear system (S3), 15 40 45 24 5 ,    To achieve higher degree of accuracy, we apply Method V-3 and obtain : [     are unknown coefficients to be determined.Solve the system of ( )

Conclusions
The system of first kind of linear Volterra integral equations has the form 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.
convergency rate of solution of the Volterra system (I2), we have Proposition 1.Given that

c c c c c c c c c c c c c c c c c c c
c c c c c c c c c c c c c c c c c c c c ,