Abstract

The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.

Keywords

Numerical Technique, Fredholm Integro-Differential Equations, Hermite Trigonometric Wavelets, Operational Matrix, Error Estimates

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1. Introduction

In this paper we solve the Fredholm Linear Integro-Differential Equations as

(1)

where, , and are given functions that have suitable derivatives, and and are given real constans. In most situations, it is difficult to obtain exact solution of the above integration. Hence various approximation method have been proposed and studied. The purpose of the present paper is to develop a trigonometric Hermite wavelet approximation for the computing of the problem [1] .

Systems of integro-differential equations have a major role in the fields of science, physical phenomena, and engineering, such as nano-hydrodynamics, glass-forming process, dropwise condensation, wind ripple in the de- sert, and modeling the competition between tumor cells and the immune system. The concept of a system of integro-differential equations has motivated a huge amount of research work in recent years. Alot of attention has been devoted to the study of differential-difference equations, e.g. equations containing shifts of the un- known function and its derivates, and also integro-differential-difference equations. For instance, see [2] [3] . There are several numerical methods for solving system of linear integro-differential equations, for example, the rationalized Haar functions method [4] , Galerkin methods with hybrid functions [5] , the spline approximation method [6] , the Chebyshev polynomial method [7] , the spectral method [8] , the CAS wavelet method [9] , Ruge- Kutta methods [10] , the Adomian decomposition methods [11] , and the interested reader can see [12] [13] for more published research works in the subject.

Our approach consists of reducing the problem to a set of linear equations by trigonometric scaling functions which is constructed for Hermite interpolation. A difficulty of using wavelet for the representation of integral operators is that quadrature leads to potentially high cost with sparse matrix. This fact particularly encourages us in efforts to devote to some appropriate wavelet bases to simplify the computation expense of the reoresentation matrix, which is importent to improve the wavelet method. Recently, the trigonometric interpolant wavelet has arisen in the approximation of operators [14] - [16] . Quack [17] has constructed a multiresolution analysis (MRA). Chen [18] [19] presented the feasibility of trigonometric wavelet numerical methods for stokes problem and Hadamard integral equation.

The organization of the rest of this paper is as follows: Section (0) describe the trigonometric scaling function on, and construct the operational matrix of derivative for these function. Section (0) summarizes the application of trigonometric scaling functions to the solution of Problem (1). Thus, a set of linear equations is formed and a solution of the considered problem is introduced. In Section (0), we report our computational results and demonstrate the accuracy of the proposed numerical schemes by presenting numerical examples. Note that we have computed the numerical results by MATLAB programming.

2. Interpolatory Hermite Trigonometric Wavelets

In this section, we will give a brief introduction of Quak’s work on the construction of Hermite interpolatory trigonometric wavelets and their basic properties. More details can be found in (see [17] ).

For all, two scaling functions and are defined as

(2)

(3)

where the Dirichlet kernel and its conjugate kernel are defined as

Obviously, , where is the linear space of trigonometric polynomials with degree not

exceeding l. The equally spaced nodes on the interval with a dyadic step are denoted by, for

any and, where is the set of all non-negative integers. Let

, for, and.

Lemma 1 (See [17] .) For we have

and their derivations are given by

Theorem 1 (Interpolatory properties of the scaling functions). (See [17] ) For, the following inter- polatory properties hold for each

(4)

From above we can take wavelet functions as scaling functions. Now, we can define the scaling function spaces. Then we have

Definition 2 (Scaling functions space). For all define the wave space as follows

As a first step of studying the spaces, the following result identifies the trigonometric polynomials which from alternative bases of these spaces.

Now a Hermite-type project operator can be introduced by means of the scaling functions. For all the Hermite projection operators mapping any real-valued differentiable -periodic function f into the space is defined as

(5)

where, , C, and are vectors with dimension. The following properties of the operators are therefore obvious:

Theorem 3 Let, and its trigonometric wavelet approximation is, then we have

where C is a positive constant value.

Proof. See [17] .

Lemma 2 (The operational matrix of scaling function derivative). (See [20] ) The differentiation of vector in 5 can be expressed as [20]

where is operational matrix of derivative for trigonometric scaling function. Suppose

(6)

where and. So the matrix can be respresented as a block matrix as

where and are matrices. The entries of matrices and may be find by using

where is a zero matrix, is a identity matrix. Using we get

(7)

Using 7 and we get

(8)

and

(9)

for.

3. Procedure Solution Using the Trigonometric Scaling Function

In this section, we first give the computational schemes for Equation (1) with the Newton-Cotes formulas. For either one of these rules, we can make a more accurate approximation by breaking up the interval into some number N of subintervals. This is called a composite rule, extended rule, or iterated rule. For example, the composite trapezoidal rule for the discretization form of (1) can be stated as

(10)

where the subintervals have the form, with and. By introducing a

basis for the subspace, the coefficients vector of the discrete solution is defined by

(11)

where C is unknown vector defined similar to (5). By substituting and using Lemma (2) in (1) we have a linear system. Now for determining unknown coefficients vector C or and, we choose collo- cation method with choosing collocation points as

(12)

Thus we have

(13)

By using Lemma 2 and after summarizing Equation (13) can be rewritten as the matrix form where, , and. Now, let us calculate the entries and in the system matrix.

where the matrix, , , and defined in Lemma (2), and I is a identity matrix. So the unknown function can be found. Note that we find these function by MATLAB.

4. Numerical Example

To support our theoretical discussion, we applied the method presented in this paper to several examples. The main objective here is to solve these two examples using the trigonometric scaling function and compare our results with exact solution.

Example 4 Consider the second-order the Fredholm Linear Integro-Differential Equation

with the mixed conditions and. The exact solution of this problem is. We

apply the suggested method with and. The behavior of the approximate solution using the proposed method with, and the exact solution are presented in Figure 1. In Table 1, we give the errors of matrix A for different values of J. From this figure, it is clear that the proposed method can be considered as an efficient method to solve the linear integral equations. From Table 1 we see the errors decrease rapidly as J increase.

In Table 2 we compare the new method with, and together with the exact solution. For the purpose of comparison we defined the meximum error for as

Example 5 Consider the following second-order the Fredholm Linear Integro-Differential Equation

with the initial conditions, and exact solution. This problem is solved by the same me- thods applied in example (4). Results are shown in Figure 2. From this figure, it is clear that the proposed method can be considered as an efficient method to solve the linear integral equations. For the purpose of com- parison in Table 3 we give the errors of matrix A for different values of J. From Table 4 we see the errors decrease rapidly as J increase. In Table 4 we compare the new method with J = 1, J = 2 and J = 3 together with the exact solution.

5. Conclusion

Our results indicate that the method with the trigonometric scaling bases can be regarded as a structurally simple algorithm that is conventionally applicable to the numerical solution of IDEs. In addition, although we have re- stricted our attention to linear Fredholm IDEs, we expect the method to be easily extended to more general IDEs. the presented method which is based on the trigonometric scaling function is proposed to find the approximate solution. A comparison of the exact solution reveals that the presented method is very effective and convenient. Nevertheless, as Figure 1 and Figure 2 illustrate, the error of the trigonometric scaling bases shows that the accuracy improves with increasing J, hence for better results, using number J is recommended. Also form the obtained approximate solution, we can conclude that the proposed method gives the solution in an excellent agree- ment with the exact solution. All computations are done using MATLAB programming.

Acknowledgements

The authors are very grateful to the editor for carefully reading the paper and for their comments and sugges- tions which have improved the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

References