Fourier Approximation for Integral Equations on the Real Line
Based on Guass quadradure method a class of integral equations having unknown periodic solution on the real line is investigated, by using Fourier series expansion for the solution of the integral equation and applying a process for changing the interval to the finite interval (; 1), the Chebychev weights become appropriate and examples indicate the high accuracy and very good approximation to the solution of the integral.
Many numerical methods for approximating the solution of integral equations are based on Galerkin and Collocation methods (see [1, 2]). Some modifications in these methods such as iterated Galerkin and iterated Collocation methods give more accurate approximations to the solution , some others have extended the technics to Petrov-Galerkin and Hammersteion equations degenerate kernel methods (see [4–7]), in some of recent methods further improvement in some steps is not possible [3, 6], the computation of sufficiently accurate solution may require the use of much partition of the domain and may thus involve the solution of a correspondingly large linear system. On the other hand, it is well known that the coefficient matrix of an integral equation is dense, and it requires heavy work to generate the coefficient matrix if the partition is much finer (see [8, 9]). To improve the difficulties in this methods, improved iterated Galerkin methods are recommended (see ), other improvement with less additional computational efforts for each step of iteration are introduced and shows that every step of iteration the coefficient matrix of the linear system to be inverted remains the same as that in the original approximation method (see ), the additional order of convergence for every step of iteration is which is less than in Galerkin, Collocation and degenerate kernel methods, where is the norm of the partition and is order of the piecewise polynomials used in the approximation. Less works consider the integral equations on the real line (see [12, 13]), integral equations with singularity (see [14, 15]), integral equations with periodic solutions (see ).
In this paper we consider the approximate solutions of Fredholm integral equations on the real line having periodic solutions.
We consider the following Fredholm integral equations of second kind with periodic solution on the real line the main part of the first section considers the several admissibility results and existence principle on real line, however before proceeding to other sections we recall some facts about periodic functions, we discuss the general approach that will be taken to obtain the various existence principles .
2. Fredholm Integral Equation on the Real Line
Let and suppose is an interval of that contains at least one compact subinterval of length , which we denote by . We define to be the subspace of (bounded continuous functions on with values in ) consisting of all -periodic mappings, that is, if , then f is continuous on and , for all t such that The norm on is the same as the norm on and it is clear that for we have We consider Fredholm integral equations over the real line that is separable where and is a polynomial of degree at most , which is nonnegative on the half line and is an arbitrary function.
Problem (2.2) can be simplified by first obtaining the partial fraction decomposition of (2.4) in the form where the sum is over all pairs of conjugate complex poles of with corresponding multiplicities Here,
Thus, without loss of generality we can consider only weights of the form In (2.2) the function is -periodic:
3. Reduction of the Integral to a Finite Interval
Milovanovic and et al.  show how to reduce the integral (2.9) to an integral on the finite interval. For this purpose we need the sum of the following series: Since (cf. ) in the simplest, but the most important case for and we obtain We bring a lemma and two theorems in the following which have been proved in .
Lemma 3.1. Let be given by (2.7), and Then
The proof of this result can be done by an integration of the function over the rectangular contour with vertices at the points where is such that the poles of the function are inside of Then, taking the corresponding integral over tends to zero, because when Then by Cauchy’s residue theorem, we get For (3.4) reduce to (3.3). When where We can suspect the following form of our sum: where is an algebraic polynomial. Indeed, we can prove the following result.
Theorem 3.2. Let and Then where is a nonnegative polynomial on of degree These polynomials satisfy the recurrence relation where
Proof. We start with (3.3) written in the form , where
Thus, the formula (3.9) is true for .
Suppose that (3.9) holds for some . Then, differentiating with respect to , we get that is, Thus, the result is proved.
We are ready now to give a transformation of the integral (2.9) to one on a finite interval. Putting and using the periodicity of the function , we have
Because of the last property of the weight function, we have
Changing the variables and putting we get the following result.
4. Gaussian Type Formula for Chebychev Weights
In order to evaluate the integral (3.20) it would seem more natural and simpler to apply the Gauss-Chebychev quadrature formula, that is, taking as in integrating function with respect to the Chebychev weight (Chw)
In this case, when for some the function satisfied the condition the error of the -point Gauss-Chebychev quadrature can be estimated as follow: where is a constant independent on and . Hence, when is very close to 1, even if the integrand is bounded, it gives a very large bound.
Thus, for evaluating the integral (3.20) it is more convenient to construct the Gaussian quadratures for the measure where the function includes the algebraic polynomial that is, Here, denotes the set of all polynomials of degree at most and is the root of Chebychev polynomials.
5. Solving Integral Equation
For getting unknown function in relation (5.1) which, is -periodic. We have the following Fourier expansion of be -periodic: indeed we will obtain unknowns coefficients and by substituting in (5.2) and using Chebychev integration the relation can be solved for these unknowns, then by substituting the Fourier coefficients in (5.3) will be determined.
6. Numerical Examples
In this section we consider some numerical examples to illustrate the presented transformation method. All computations were done by Maple 11, and the runtime for this calculation is too short. It should be noted that the real line is the integration interval of the following examples, but the integrands are periodic functions and they have been evaluated in the period .
Example 6.1. Let us consider the following integral equation: where and are arbitrary knowns and is unknown -periodic function on which should be determined. Also and are equal 1 in (2.8).Applying Gaussian quadrature with the ChW for , we get the approximate Fourier expantion of with relative error given in Table 1. The exact solution of (6.1) is the approximated solution of according to the values of and is which shows the difference of the solutions for equals and other values of the errors are given in Table 1, which indicates a super convergence of the solution even for small values of .
Example 6.2. Consider the integral equation where and are arbitrary and is -periodic function on also and are equal 1 in (2.8).
Applying Gaussian quadrature with the Chw, for .
The exact and approximated solutions are considered to illustrate in Table 2 with comparison is exact solution the for the error is almost zero.
Example 6.3. Consider the following integral equation: where is arbitrary amount and is any function of . If and by getting (Chw) quadrature we will obtain the approximations function for . The exact solution is if then and for and So the approximation and exact solution almost equal, or The approximation solution is almost exact.
A Fourier series approximation for the integral equations having periodic solution are considered in the real line, the procedure produced almost an exact approximation for the solution of the integral even by taking few number of the elements in the Fourier series expansion of the solution, the numerical examples indicate the high accuracy of the approximation.
R. Kress, Linear Integral Equations, vol. 82 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1989.View at: MathSciNet