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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.

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 [

Since the paper [

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

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 [

The lowest degree continuous cardinal spline is

The cardinal spline

Then

construction,

To achieve higher degree of smoothness, we employ

It is a simple calculation to check the cardinal condition is satisfied. The

ones for

In this and next two sections, we are concentrating on the second kind system of linear Volterra integral equations

where

As for the Volterra system (I2), we solve it in an interval

in (I2), we get

Let

which is a simple system of

Proposition 1. Given that

more

where

where

The proof is very similar to the proof of the following Proposition 3, so we skip it.

To achieve higher approximation rate, we plug

into (I2), where extra function values still follow the conditions (cond1) and arrive at

Let

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

approximates the solution of the system (I2) with a rate of

Proposition 2. Given that

where

where

The proof is very similar to the proof of the following Proposition 3, so we skip it.

To achieve higher approximation rate, we let

again we plug into (I2), where extra function values still follow the conditions (cond2) cf [

Let

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

Proposition 3. Given that

thermore,

where

Let

where

Proof. Let

where the coefficients are the solutions of above linear system (S3), and

Plug in

Therefore

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

and

Since the solution is ^{ }for

Example 3. Consider the third order differential equations:

with the initial condition:

Let

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

The system of first kind of linear Volterra integral equations has the form

where

They can easily be transformed to the system of second kind of linear Fredholm and Volterra integral equations (cf [

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.

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.

XiaoyanLiu,ZhiLiu,JinXie, (2015) Solving Systems of Volterra Integral Equations with Cardinal Splines. Journal of Applied Mathematics and Physics,03,1422-1430. doi: 10.4236/jamp.2015.311170