Fourier Approximation for Integral Equations on the Real Line

Abstract

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; 1), the Chebychev weights become appropriate and examples indicate the high accuracy and very good approximation to the solution of the integral.

1. Introduction

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 [3], 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 [10]), 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 [11]), the additional order of convergence for every step of iteration is ℎ^4𝑟, which is less than ℎ^2𝑟, in Galerkin, Collocation and degenerate kernel methods, where ℎ is the norm of the partition and 𝑟 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 [16]).

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 𝑓(𝑥)=𝑔(𝑥)+𝜆_ℝ𝑘(𝑥,𝑡)𝑓(𝑡)𝑑𝑡, (1.1) 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 [16].

2. Fredholm Integral Equation on the Real Line

Let 0<𝑇<1 and suppose 𝐼 is an interval of ℝ that contains at least one compact subinterval of length 𝑇, which we denote by 𝐼_𝑇. We define 𝐴_𝑇(𝐼) to be the subspace of 𝐵𝐶(𝐼) (bounded continuous functions on 𝐼 with values in ℝ) consisting of all 𝑇-periodic mappings, that is, if 𝑓∈𝐴_𝑇(𝐼), then f is continuous on 𝐼 and 𝑓(𝑡+𝑇)=𝑓(𝑡), for all t such that 𝑡+𝑇∈𝐼. The norm on 𝐴_𝑇(𝐼) is the same as the norm ‖.‖₀ on 𝐵𝐶(𝐼), and it is clear that for 𝑓∈𝐴_𝑇(𝐼) we have ‖𝑓‖₀=sup_𝑡∈𝐼|𝑦(𝑡)|=sup_{𝑡∈𝐼𝑇}|𝑦(𝑡)|. (2.1) We consider Fredholm integral equations over the real line ℝ𝑓(𝑥)=𝑔(𝑥)+𝜆_ℝ𝑘(𝑥,𝑡)𝑓(𝑡)𝑑𝑡, (2.2) that 𝑘(𝑥,𝑡) is separable 𝑘(𝑥,𝑡)=^𝑛_𝑖=1𝛾_𝑖(𝑥)𝜔_𝑖𝜔(𝑡),_𝑖𝑃(𝑡)=^𝑖(𝑡²)𝑄_𝑖(𝑡²),𝑖=1,2…,𝑛_𝑖, (2.3)(2.4) where 𝑄_𝑖(𝑡)=^𝑛𝑖_𝑘=1(𝑡+𝑏²_𝑘),0<𝑏₁≤𝑏₂≤⋯≤𝑏_𝑛𝑖, (2.5) and 𝑃_𝑖(𝑡) is a polynomial of degree at most 𝑚_𝑖<𝑛_𝑖, which is nonnegative on the half line [0,+∞), and 𝛾_𝑖(𝑥) is an arbitrary function.

Problem (2.2) can be simplified by first obtaining the partial fraction decomposition of (2.4) in the form 𝜔_𝑖(𝑡)=^𝑚𝑖𝑟_𝑗=1^𝑗_𝜈𝑖=1𝑐_𝑗𝜈𝑖(𝑡²+𝑏²_𝑗)², (2.6) where the sum is over all pairs of conjugate complex poles ±𝑖𝑏_𝑗, of 𝑄(𝑡²), with corresponding multiplicities 𝑟_𝑗,𝑗=1,…,𝑚_𝑖. Here, ∑^𝑚𝑖_𝑗=1𝑟_𝑗=𝑛.

Thus, without loss of generality we can consider only weights of the form 𝜔_𝜈𝑖(𝑡)=𝜔_𝜈𝑖1(𝑡;𝑏)=(𝑡²+𝑏²)^𝜈𝑖,𝜈_𝑖≥1. (2.7) In (2.2) the function 𝑓(𝑥), is 2𝜋 -periodic:

𝑓(𝑥)=𝑔(𝑥)+𝜆_ℝ^𝑛_𝑖=1𝛾_𝑖(𝑥)𝜔_𝜈𝑖(𝑡)𝑓(𝑡)𝑑𝑡=𝑔(𝑥)+𝜆^𝑛_𝑖=1𝛾_𝑖(𝑥)_ℝ𝜔_𝜈𝑖(𝑡)𝑓(𝑡)𝑑𝑡=𝑔(𝑥)+𝜆^𝑛_𝑖=1𝛾_𝑖(𝑥)_ℝ𝑓(𝑡)(𝑡²+𝑏²)^𝜈_𝑖𝑑𝑡. (2.8)

Let 𝐼_𝜈𝑖(𝑓)=𝐼_𝜈𝑖(𝑓;𝑏)=_ℝ𝑓(𝑡)𝑑𝑡(𝑡²+𝑏²)^𝜈𝑖,𝑏>0,𝜈_𝑖≥1. (2.9)

3. Reduction of the Integral to a Finite Interval

Milovanovic and et al. [17] 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: 𝑊_𝜈𝑖(𝑡)=𝑊_𝜈𝑖(𝑡;𝑏)=^∞_𝑘=−∞𝑤_𝜈𝑖(2𝑘𝜋+𝜏)=^∞_𝑘=−∞1[(2𝑘𝜋+𝜏)²+𝑏²]^𝜈𝑖. (3.1) Since (cf. [9]) ^∞_𝑘=−∞1(𝑘+𝛼)²+𝛽²=𝜋𝛽sinh2𝜋𝛽,cosh2𝜋𝛽−cos2𝜋𝛼 (3.2) in the simplest, but the most important case 𝜈_𝑖=1, for 2𝜋𝛼=𝜏 and 2𝜋𝛽=𝑏, we obtain 𝑊₁(𝜏)=𝑊₁(𝑡;𝑏)=sinh𝑏12𝑏.cosh𝑏−cos𝜏 (3.3) We bring a lemma and two theorems in the following which have been proved in [17].

Lemma 3.1. Let 𝑤_𝜈𝑖 be given by (2.7), 𝜁^±=−(𝜏±𝑖𝑏)/2𝜋 and 𝜁=−𝜁⁺. Then 𝑊_𝜈𝑖(𝜏)=−(2𝜋)^{1−2𝜈𝑖}2(𝜈_𝑖−1)!{lim_𝑧→𝜁+𝑑^𝜈𝑖−1𝑑𝑧^𝜈𝑖−1[cot𝜋𝑧(𝑧+𝜁)^𝜈𝑖]+lim_{𝑧→𝜁−}𝑑^𝜈𝑖−1𝑑𝑧^𝜈𝑖−1[cot𝜋𝑧(𝑧+𝜁)^𝜈𝑖]}. (3.4)

The proof of this result can be done by an integration of the function 𝑧↦𝑔(𝑧)=𝜋cot(𝜋𝑧)𝑤_𝜈𝑖(2𝜋𝑧+𝜏) over the rectangular contour 𝐶_𝑁 with vertices at the points (𝑁+(1/2))(±1±𝑖,) where 𝑁∈ℕ is such that the poles 𝜁^± of the function 𝑔 are inside of 𝐶_𝑁. Then, taking 𝑁→+∞, the corresponding integral over 𝐶_𝑁 tends to zero, because 𝑤_𝜈𝑖(𝑧)=𝐎(1/𝑧^2𝜈𝑖) when 𝑧→∞. Then by Cauchy’s residue theorem, we get 𝑊_𝜈𝑖(𝜏)=^∞_𝑘=−∞𝑤_𝜈𝑖(2𝑘𝜋+𝜏)=−(Res𝑔(𝑧)_𝑧=𝜁⁺+Res𝑔(𝑧)_𝑧=𝜁⁻). (3.5) For 𝜈_𝑖=1, (3.4) reduce to (3.3). When 𝑊_𝜈𝑖(𝜏)=𝑏cosh𝑏−sinh𝑏4𝑏³⋅cos𝜏+𝑎(cosh𝑏−cos𝜏)², (3.6) where 𝑎=sinh2𝑏−2𝑏.2𝑏cosh𝑏−2sinh𝑏 (3.7) We can suspect the following form of our sum: 𝑊_𝜈𝑖=𝑝_𝜈𝑖(cos𝜏)(cosh𝑏−cos𝜏)², (3.8) where 𝑝_𝜈𝑖(𝑥), is an algebraic polynomial. Indeed, we can prove the following result.

Theorem 3.2. Let 𝑥=cos𝜏 and 𝑐=cosh𝑏. Then 𝑊_𝜈𝑖(𝜏)=𝑊_𝜈𝑖𝑝(𝜏;𝑏)=^𝜈𝑖(𝑥)(𝑐−𝑥)^𝜈𝑖,𝜈_𝑖=1,2…,𝑖=1,2…, (3.9) where 𝑝_𝜈𝑖(𝑥)=𝑝_𝜈𝑖(𝑥;𝑏) is a nonnegative polynomial on [−1,1] of degree 𝜈_𝑖−1. These polynomials satisfy the recurrence relation 𝑝_𝜈𝑖+11(𝑥)=2𝑏𝜈_𝑖{𝜈_𝑖√𝑐²−1𝑝_𝜈𝑖−(𝑐−𝑥)𝜕𝑝_𝜈𝑖(𝑥)𝜕𝑏}, (3.10) where 𝑝₁√(𝑥)=𝑐²−12𝑏. (3.11)

Proof. We start with (3.3) written in the form (𝑐−𝑥)𝑊₁(𝜏)=𝑝₁(𝑥), where 𝑝₁(𝑥)=sinh𝑏=√2𝑏𝑐²−12𝑏,𝑥=cos𝜏,𝑐=cosh𝑏. (3.12) Thus, the formula (3.9) is true for 𝜈=1.
Suppose that (3.9) holds for some (𝜈≥1). Then, differentiating (𝑐−𝑥)^𝜈𝑊_𝜈(𝜏)=𝑝_𝜈(𝑥) (3.13) with respect to 𝑏, we get 𝜈(𝑐−𝑥)^𝜈−1𝑑𝑐𝑊𝑑𝑏^𝜈(𝜏)+(𝑐−𝑥)^𝜈=𝜕𝑝_𝜈(𝑥),𝜕𝑏 (3.14) that is, (𝑐−𝑥)^𝜈+1𝑊_𝜈+11(𝜏)=2𝑏𝜈{𝜈(𝑐²−1)𝑝_𝜈(𝑥)−(𝑐−𝑥)𝜕𝑝_𝜈(𝑥)𝜕𝑏}=𝑝_𝜈+1(𝑥). (3.15) 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 𝑡=2𝑘𝜋+𝜏 and using the periodicity of the function 𝑓, 𝑓(𝑡)=𝑓(2𝑘𝜋+𝜏)=𝑓(𝜏), (3.16) we have

𝐼_𝜈𝑖(𝑓)=𝐼_𝜈𝑖(𝑓;𝑏)=^∞_𝑘=−∞^{(2𝑘+1)𝜋}_{(2𝑘−1)𝜋}𝑓(𝑡)𝑤_𝜈𝑖=(𝑡)𝑑𝑡_{∞𝑘=−∞+𝜋−𝜋}𝑓(𝜏)𝑤_𝜈𝑖=(2𝑘𝜋+𝜏)𝑑𝜏_𝜋−𝜋𝑓(𝜏)(^∞_𝑘=−∞𝑤_𝜈𝑖(2𝑘𝜋+𝜏))𝑑𝜏, (3.17)

because of the uniform convergence of the series (3.1). Thus, 𝐼_𝜈𝑖(𝑓)=^+𝜋_−𝜋𝑓(𝜏)𝑤_𝜈𝑖(𝜏)𝑑𝜏, (3.18) where 𝑤_𝜈𝑖(𝜏) is defined by (3.1) and given by Theorem (3.2). We see that 𝑤_𝜈𝑖(−𝜏)=𝑤_𝜈𝑖(𝜏), that is, 𝑤_𝜈𝑖 is an even weight function.

Because of the last property of the weight function, we have 𝐼_𝜈𝑖(𝑓)=𝐼_𝜈𝑖(𝑓;𝑏)=⁰_−𝜋𝑓(𝜏)𝑤_𝜈𝑖(𝜏)𝑑𝜏+^𝜋₀𝑓(𝜏)𝑤_𝜈𝑖(𝜏)𝑑𝜏=^𝜋₀(𝑓(𝜏)+𝑓(−𝜏))𝑤_𝜈𝑖(𝜏)𝑑𝜏. (3.19)

Changing the variables cos𝜏=𝑥 and putting 𝑓(𝜏)+𝑓(−𝜏)=𝐹(cos𝜏), (3.20) we get the following result.

Theorem 3.3. The integral (2.9) can be transformed to the form 𝐼_𝜈𝑖(𝑓)=𝐼_𝜈𝑖(𝑓;𝑏)=¹₋₁𝑝𝐹(𝑥)^𝜈𝑖(𝑥)(𝑐−𝑥)^𝜈𝑖.𝑑𝑥√1−𝑥², (3.21) where 𝑐=cosh𝑏,𝑝_𝜈𝑖(𝑥) is a polynomial determined by the recurrence relation Theorem 3.2, and 𝐹 is defined by (3.20).

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 𝑥↦𝜙(𝑥)=𝐹(𝑥)𝑝_𝜈𝑖(𝑥)/(𝑐−𝑥)^𝜈𝑖,𝑐>1, as in integrating function with respect to the Chebychev weight (Chw) 𝜐₀(𝑥)=(1−𝑥²)^−1/2.

In this case, when for some 𝑟≥1 the function 𝐹 satisfied the condition ¹₋₁𝐹^(𝑟)(𝑥)(√1−𝑥²)^𝑟−1𝑑𝑥<+∞, (4.1) the error 𝐑_𝑛(𝜙)_𝜈0 of the 𝑛-point Gauss-Chebychev quadrature can be estimated as follow: |𝐑_𝑛(𝜙)_𝜈0𝐴|≤𝑛^𝑟1₋₁|𝑑^𝑟𝑑𝑥^𝑟[𝐹(𝑥)𝑝_𝜈𝑖(𝑥)(𝑐−𝑥)^𝜈𝑖]|(1−𝑥²)^(𝑟−1)/2𝑑𝑥, (4.2) where 𝐴>0 is a constant independent on 𝜙 and 𝑛. Hence, when 𝑐>1, 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 ¹₋₁𝜙(𝑥)𝑑𝜆_𝜈𝑖𝜋(𝑥)=𝑛^𝑛_𝑘=1𝜙(𝑥_𝑖)+𝐑_𝑛(𝜙)_𝜈0,(𝐑_𝑛(𝑝_2𝑛−1)_𝜈0)≡0, (4.3) for the measure 𝑑𝜆_𝜈𝑖(𝑥)=𝑑𝑥√1−𝑥², (4.4) where the function 𝜙 includes the algebraic polynomial 𝑝_𝜈𝑖(𝑥), that is, 𝜙(𝑥)=𝐹(𝑥)𝑝_𝜈𝑖(𝑥)/(𝑐−𝑥)^𝜈𝑖. Here, 𝑝_2𝑛−1 denotes the set of all polynomials of degree at most 2𝑛−1 and 𝑥_𝑖 is the root of Chebychev polynomials.

5. Solving Integral Equation

Let this integral equation 𝑓(𝑥)=𝑔(𝑥)+𝜆_𝑅𝑘(𝑥,𝑡)𝑓(𝑡)𝑑𝑡, (5.1) which 𝑓(𝑡) is 2𝜋-periodic, 𝑘(𝑥,𝑡), BE separable, that this integral equation is simplified according to Sections 2 and 3: 𝑓(𝑥)=𝑔(𝑥)+𝜆^𝑛_𝑖=1𝛾_𝑖(𝑥)¹₋₁𝑝𝐹(𝑡)⋅^𝜈𝑖(𝑡)(𝑐−𝑥)^𝜈𝑖⋅𝑑𝑡√1+𝑡²𝑑𝑡, (5.2) where 𝑓(𝜏)+𝑓(−𝜏)=𝐹(cos𝑥)=𝐹(𝑥),𝑐=cosh𝑏 and 𝑝_𝜈𝑖(𝑥) is obtained from Theorem 3.2.

For getting unknown function 𝑓(𝑥) in relation (5.1) which, 𝑓(𝑥) is 2𝜋-periodic. We have the following Fourier expansion of 𝑓(𝑥) be 2𝜋-periodic: 𝑎𝑓(𝑥)≃⁰2+^𝑚_𝑘=1(𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥), (5.3) indeed 𝑎𝐹(cos𝜏)=𝐹(𝑥)=𝑓(𝜏)+𝑓(−𝜏)=⁰2+2^𝑚_𝑘=1(𝑎_𝑘cos𝑘𝑥), (5.4) we will obtain 2𝑚+1 unknowns coefficients 𝑎₀,𝑎_𝑘 and 𝑏_𝑘,𝑘=1…𝑚, by substituting in (5.2) and using Chebychev integration the relation can be solved for these unknowns, then by substituting the Fourier coefficients 𝑓(𝑥), 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 (−1,1).

Example 6.1. Let us consider the following integral equation: 𝜋𝑓(𝑥)=cos(𝑥)−𝑒𝜆𝛾(𝑥)+𝜆𝛾(𝑥)_ℝ𝑓(𝑡)1+𝑡²𝑑𝑡, (6.1) where 𝜆 and 𝛾(𝑥) are arbitrary knowns and 𝑓(𝑥) is unknown 2𝜋-periodic function on ℝ, which should be determined. Also 𝜈_𝑖 and 𝑛 are equal 1 in (2.8).Applying Gaussian quadrature with the ChW for 𝑚=5,6,7,8,9, we get the approximate Fourier expantion of 𝑓(𝑥) with relative error given in Table 1. The exact solution of (6.1) is cos(𝑥), the approximated solution of 𝑓(𝑥) according to the values of 𝑎₀=(0.4111077167𝑒−3), and 𝑎₁=1,𝑎₂=𝑎₃=𝑎₄=𝑎₅=𝑏₁=𝑏₂=𝑏₃=𝑏₄=𝑏₅=0, is ℎ(𝑥)=0.4111077167𝑒−3/2+cos(𝑥) which shows the difference of the solutions for 𝑚=5 equals 𝑎₀/2, and other values of 𝑚 the errors are given in Table 1, which indicates a super convergence of the solution even for small values of 𝑚.

Table 1 

Example 6.2. Consider the integral equation 𝑓(𝑥)=sin(𝑥)1+cos(𝑥)+𝜆𝛾(𝑥)_ℝ𝑓(𝑡)1+𝑡²𝑑𝑡, (6.2) where 𝜆 and 𝛾(𝑥) are arbitrary and 𝑓(𝑥) is 2𝜋-periodic function on ℝ, also 𝜈_𝑖 and 𝑚 are equal 1 in (2.8).

Applying Gaussian quadrature with the Chw, for 𝑚=5,6,8.

The exact and approximated solutions are considered to illustrate in Table 2 with comparison is exact solution the 𝑓(𝑥)=sin(𝑥)/(1+cos(𝑥)), for 𝑚=8 the error is almost zero.

Table 2 

Example 6.3. Consider the following integral equation: 𝑓(𝑥)=sin(𝑥)+𝜆𝛾(𝑥)_ℝ𝑓(𝑡)(1+𝑡²)²𝑑𝑡, (6.3) where 𝜆 is arbitrary amount and 𝛾(𝑥) is any function of 𝑥. If 𝜈_𝑖=2 and 𝑛=1, by getting (Chw) quadrature we will obtain the approximations function for 𝑓(𝑥). The exact solution is 𝑓(𝑥)=sin(𝑥), if 𝑚=5 then 𝑏₁=1 and 𝑎_𝑖=𝑏_𝑗=0, for 𝑖=1,…,5, and,𝑗=2,…,5. So the approximation and exact solution almost equal, or 𝑓(𝑥)=ℎ(𝑥). The approximation solution is almost exact.

7. Conclusion

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.