#### Abstract

The cardinal trigonometric splines on small compact supports are employed to solve integral equations. The unknown function is expressed as a linear combination of cardinal trigonometric splines functions. Then a simple system of equations on the coefficients is deducted. When solving the Volterra integral equations, the system is triangular, so it is relatively straight forward to solve the nonlinear system of the coefficients and a good approximation of the original solution is obtained. The sufficient condition for the existence of the solution is discussed and the convergence rate is investigated.

#### 1. Introduction

Trigonometric splines were introduced by Schoenberg in [1]. Univariate trigonometric splines are piecewise trigonometric polynomials of the form (where are real numbers) in each interval and they are nature extensions of polynomial splines. Needless to say, trigonometric splines have their own advantages. A number of papers have appeared to study the properties of the trigonometric splines and trigonometric B-splines (cf. [2–4]) since then.

In my previous papers (cf. [5–7]), low degree orthonormal spline and cardinal spline functions with small compact supports were constructed. The method can be extended to construct higher degree orthonormal or cardinal splines. Unlike in the book (cf. [1]), by the cardinal splines we mean the specific splines satisfying cardinal interpolation conditions, which means that the cardinal function has the value one at one interpolation point and value zero at all other interpolation points. Cardinal splines are not only useful in interpolation problems, but they are also useful in deduction of numerical integration formulas [6] and in solving integral equations.

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, and geophysics. 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.

In this paper we focus on the Volterra integral equations of the second kind: where is a complex number, the kernel , , and are known functions, and is an unknown function to be determined.

This paper has six sections. In Section 2, a univariate trigonometric cardinal spline on a small compact support is constructed and properties are studied. In Section 3, the applications of trigonometric cardinal splines on solving the Volterra integral equations are explored. The unknown function is expressed as a linear combination of trigonometric cardinal spline functions. Then a simple system of nonlinear equations on the coefficients is deducted. It is relatively simple to solve the linear system since the system is triangular, and a good approximation of the original solution is obtained. The sufficient condition for the existence is discussed and the convergence rate is investigated. In Section 4, the applications of trigonometric cardinal splines on solving the systems of Volterra integral equations are explored. In Section 5, numerical examples are given on solving the nonlinear Volterra integral equations and a system of nonlinear Volterra integral equations. Section 6 contains the conclusion remarks.

#### 2. A Cardinal Trigonometric Spline with a Small Support

To construct cardinal trigonometric splines, let

This is the zero degree polynomial or trigonometric B-spline.

Let . A continuous univariate cardinal trigonometric spline with a small support is

Explicitly, The graph of is Figure 1.

Proposition 1. *If , exists and is bounded on the finite interval (where ), for any and any integer , such that ; let
**
then .**If , exists on and both and are bounded, for any and any chosen , let
**
then .*

#### 3. Numerical Method Solving Integral Equations

To solve the Volterra integral equations (2) in an interval , we let , , . Furthermore, let plugging in (2), we get

Letting , we arrive for , (where , ,) at

which is a triangular system of nonlinear equations on unknowns . Notice that the coefficient matrix for the system is a triangular matrix, which means that we solve , where is a number not depending on , for . For the convergence rate of solution of the Volterra integral equations (2), we have the following Proposition 2.

Proposition 2. *Given that , , , and exist and are bounded in , , () exist and are bounded in . Furthermore, satisfies the condition
**
where . Let be an integer, , let , , , and satisfies the linear system **
then
**
where is the exact solution of (2).*

*Proof. *Let
where the coefficients are the solutions of above system . Then

Plug in
Therefore

#### 4. Numerical Method Solving Systems of Integral Equations

The system of Volterra integral equations is critical to many physical, biological, and engineering models. For instance, for some heat transfer problems in physics, the heat equations are usually replaced by a system of Volterra integral equations [8]. Many well-known models for neural networks in biomathematics, nuclear reactor dynamics problems, and thermoelasticity problems are also based on a system of Volterra integral equations ([9, 10]). Our method could be extended to solve the system of Volterra integral equations. Given in an interval , we let , , . Furthermore, let () plugging in (17), we get

Let , we arrive for , (where , ,) at which is a triangular system of nonlinear equations on unknowns . Notice that the coefficient matrix for the system is a triangular matrix, which means that we solve , where is a number, for . For the convergence rate of solution of the Volterra integral equations (2), we have the following Proposition 3.

Proposition 3. *Given that , , , and exists and is bounded in , , exist and are bounded in . Furthermore, satisfies the condition
**
where . Let be an integer, , let , , , satisfies the linear system **
then
**
where is the exact solution of (17).*

#### 5. Numerical Examples

*Example 1. *Given that , where , , .

Let , , , . , , , . Plugging into the integral equation, we arrive at

The solution is [, , , , , , , , , , ] = [0.4162361433, 0.5472481952, 0.6478042689, 0.7326106099, 0.8073700605, 0.8750164888, 0.9373063268, 0.9954057887, 1.050163033, 1.102269297].

Compared with the exact solution [0.4162277660, 0.5472135955, 0.6477225575, 0.7324555320, 0.8071067811, 0.8745966692, 0.9366600265, 0.9944271910, 1.048683298, 1.1]. The error .

*Example 2. *, , with the exact solution .

We choose , , and , and let plug into . The system has the form

The solution is [, , , , , , , , , , ] = [−0.5463024898, −0.4228999647, −0.3095798686, −0.2031136045, −0.1009105373, −0.0007472398948, 0.09943058913, 0.201675057, 0.308203618, 0.4215987172, 0.5450804126]. Comparing with the exact solutions , , , , , , , , , , = [−0.5463024898, −0.4227932187, −0.3093362496, −0.2027100355, −0.1003346720, 0, 0.1003346720, 0.2027100355, 0.3093362496, 0.4227932187, 0.5463024898], the error .

Notice that our accuracy is much better than the one in the paper [11] although they choose in the paper.

*Example 3. *The equation of percolation in [12]
where physical parameter, constant ; according to paper [13], we can obtain a unique nonnegative nontrivial solution. Let , , , . , , . Let , ; plugging into the integral equation, we arrive at

The result we got is [] = [0, 0.003026, 0.013703, 0.035223, 0.071853, 0.129210, 0.214645, 0.337732, 0.510886, 0.750148, 1.076178].

We do not have the exact solution. Nevertheless, compare with = , [, , , , , , , , , , ] = [0, 0.003011, 0.013632, 0.035039, 0.071478, 0.128539, 0.213538, 0.336002, 0.508289, 0.746365, 1.070795].

For the same integral equation, let , ; we arrive at [, , , , , , , , , , ] = [0.012903, 0.051423, 0.116203, 0.213289, 0.351521, 0.542618, 0.801744,1.148333, 1.607165, 2.209747].

Compare with , [, , , , , , , , , , ] = [0.012840, 0.051156, 0.115595, 0.212177, 0.349700, 0.539832, 0.797666, 1.142548, 1.599148, 2.198834].

*Example 4. *Consider the equation = .

The exact solution is .

Applying the method on the interval , let , , , , , and ; plugging into the integral equation (2), we arrive at

The solution is [] = [0.1001681505, 0.2013552395, 0.3046865884, 0.4115060574, 0.5235819595, 0.6434770162, 0.775364957, 0.9272531678, 1.119717038, 1.570732818].

Compared with the exact solution: , , , , , , , , , , ] = [0.1001674211, 0.2013579207, 0.3046926540, 0.4115168460, 0.5235987755, 0.6435011087, 0.7753974965, 0.9272952180, 1.119769515, 1.570796326]. The error .

*Example 5. *Consider the equation = = ; the exact solution is .

Applying the method on the interval , let , , , , , and ; plugging into the integral equation, we arrive at

The solution is [, , , , , , , , , , ] = [0.09531139, 0.18232333, 0.2623661939, 0.3364740924, 0.4054667723, 0.4700050464, 0.5306294034, 0.5877875548, 0.6418545277, 0.6931475935].

Compared with the exact solution [, , , , , , , , , , ] = [0.09531017980, 0.1823215567, 0.2623642644, 0.3364722366, 0.4054651081, 0.4700036292, 0.5306282510, 0.5877866649, 0.6418538861, 0.6931471805]. The error .

*Example 6. *Consider a system of Volterra integral equations:

The exact solutions are , .

In an interval , we let , , . Furthermore, let
plugging into the system, we get
which is a nearly triangular system of nonlinear equations on unknowns , ; we need to solve two nonlinear equations: , , where are numbers, each time for .

Solutions are , = 0.09983172966, 0.1986591558, 0.2954877407, 0.3893423816, 0.4792769003, 0.5643832656, 0.6438004752, 0.7167230387, 0.782409006, 0.8405334929, 0.9949873745, 0.9799992481, 0.9551843698, 0.9207890215, 0.8771545950, 0.8247142533, 0.7639887345, 0.6955813657, 0.6201723633, 0.5443657395. Compared with the exact solution: 0, 0.09983341664, 0.1986693307, 0.2955202066, 0.3894183423, 0.4794255386, 0.5646424733, 0.6442176872, 0.7173560908, 0.7833269096, 0.8414709848, 1, 0.9950041652, 0.9800665778, 0.9553364891, 0.9210609940, 0.8775825618, 0.8253356149, 0.7648421872, 0.6967067093, 0.6216099682, 0.5403023058. The error <0.005.

#### 6. Conclusions

The proposed method is a simple and effective procedure for solving nonlinear Volterra integral equations, as well as a system of nonlinear Volterra integral equations. The methods can be adapted easily to the Volterra integral equations of the first kind, which have the form . The convergence rate could be higher if we use more complicated orthonormal or cardinal splines. Nevertheless, the resulting system of coefficients will be more complicated nonlinear systems, which could take more time and effort to solve.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was funded by the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15 and the Major Project of the Nature Science Foundation of the Education Department, Anhui Province, under Grant no. KJ2014ZD30.