Abstract and Applied Analysis

Volume 2013 (2013), Article ID 426916, 17 pages

http://dx.doi.org/10.1155/2013/426916

Research Article

Numerical Methods for Solving Fredholm Integral Equations of Second Kind

S. Saha Ray and P. K. Sahu

Department of Mathematics, National Institute of Technology, Rourkela 769008, India

Received 3 September 2013; Accepted 3 October 2013

Academic Editor: Rasajit Bera

Copyright © 2013 S. Saha Ray and P. K. Sahu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

1. Introduction

Integral equations occur naturally in many fields of science and engineering [1]. A computational approach to solve integral equation is an essential work in scientific research.

Integral equation is encountered in a variety of applications in many fields including continuum mechanics, potential theory, geophysics, electricity and magnetism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control systems, communication theory, mathematical economics, population genetics, queuing theory, medicine, mathematical problems of radiative equilibrium, the particle transport problems of astrophysics and reactor theory, acoustics, fluid mechanics, steady state heat conduction, fracture mechanics, and radiative heat transfer problems. Fredholm integral equation is one of the most important integral equations.

Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed.

A computational approach to solving integral equation is an essential work in scientific research. Some methods for solving second kind Fredholm integral equation are available in the open literature. The B-spline wavelet method, the method of moments based on B-spline wavelets by Maleknejad and Sahlan [2], and variational iteration method (VIM) by He [3–5] have been applied to solve second kind Fredholm linear integral equations. The learned researchers Maleknejad et al. proposed some numerical methods for solving linear Fredholm integral equations system of second kind using Rationalized Haar functions method, Block-Pulse functions, and Taylor series expansion method [6–8]. Haar wavelet method with operational matrices of integration [9] has been applied to solve system of linear Fredholm integral equations of second kind. Quadrature method [10], B-spline wavelet method [11], wavelet Galerkin method [12], and also VIM [13] can be applied to solve nonlinear Fredholm integral equation of second kind. Some iterative methods like Homotopy perturbation method (HPM) [14–16] and Adomian decomposition method (ADM) [16–18] have been applied to solve nonlinear Fredholm integral equation of second kind.

2. Fredholm Integral Equation

The general form of linear Fredholm integral equation is defined as follows:

where and are both constants. , , and are known functions while is unknown function. (nonzero parameter) is called eigenvalue of the integral equation. The function is known as kernel of the integral equation.

2.1. Fredholm Integral Equation of First Kind

The linear integral equation is of form (by setting in (1))

Equation (2) is known as Fredholm integral equation of first kind.

2.2. Fredholm Integral Equation of Second Kind

The linear integral equation is of form (by setting in (1))

Equation (3) is known as Fredholm integral equation of second kind.

2.3. System of Linear Fredholm Integral Equations

The general form of system of linear Fredholm integral equations of second kind is defined as follows:

where and are known functions and are the unknown functions for .

2.4. Nonlinear Fredholm-Hammerstein Integral Equation of Second Kind

Nonlinear Fredholm-Hammerstein integral equation of second kind is defined as follows:

where is the kernel of the integral equation, and are known functions, and is the unknown function that is to be determined.

2.5. System of Nonlinear Fredholm Integral Equations

System of nonlinear Fredholm integral equations of second kind is defined as follows:

where and are known functions and are the unknown functions for .

3. Numerical Methods for Linear Fredholm Integral Equation of Second Kind

Consider the following Fredholm integral equation of second kind defined in (3)

where and are known functions and is unknown function to be determined.

3.1. B-Spline Wavelet Method

3.1.1. B-Spline Scaling and Wavelet Functions on the Interval

Semiorthogonal wavelets using B-spline are specially constructed for the bounded interval and this wavelet can be represented in a closed form. This provides a compact support. Semiorthogonal wavelets form the basis in the space .

Using this basis, an arbitrary function in can be expressed as the wavelet series. For the finite interval , the wavelet series cannot be completely presented by using this basis. This is because supports of some basis are truncated at the left or right end points of the interval. Hence, a special basis has to be introduced into the wavelet expansion on the finite interval. These functions are referred to as the boundary scaling functions and boundary wavelet functions.

Let and be two positive integers and let

be an equally spaced knots sequence. The functions

are called cardinal B-spline functions of order for the knot sequence and .

By considering the interval , at any level , the discretization step is , and this generates number of segments in with knot sequence

Let be the level for which ; for each level, , the scaling functions of order can be defined as follows in [2]:

And the two scale relations for the -order semiorthogonal compactly supported B-wavelet functions are defined as follows:

where .

Hence, there are boundary wavelets and inner wavelets in the bounded interval . Finally, by considering the level with , the B-wavelet functions in can be expressed as follows:

The scaling functions occupy segments and the wavelet functions occupy segments.

When the semiorthogonal wavelets are constructed from B-spline of order , the lowest octave level is determined in [19, 20] by

so as to have a minimum of one complete wavelet on the interval .

3.1.2. Function Approximation

A function defined over may be approximated by B-spline wavelets as [21, 22]

If the infinite series in (15) is truncated at , then (15) can be written as [2]

where and are scaling and wavelets functions, respectively, and and are vectors given by

with

where and are dual functions of and , respectively. These can be obtained by linear combinations of , , and , , , as follows. Let

Using (11), (20), (12)-(13), and (21), we get

Suppose that and are the dual functions of and , respectively; then

3.1.3. Application of B-Spline Wavelet Method

In this section, linear Fredholm integral equation of the second kind of form (7) has been solved by using B-spline wavelets. For this, we use (16) to approximate as

where is defined in (18) and is unknown vector defined similarly as in (17). We also expand and by B-spline dual wavelets defined in (24) as

where

From (26) and (25), we get

since

By applying (25)–(28) in (7) we have

By multiplying both sides of (30) with from the right and integrating both sides with respect to from 0 to 1, we get

since

and is a square matrix given by

Consequently, from (31), we get . Hence, we can calculate the solution for .

3.2. Method of Moments

3.2.1. Multiresolution Analysis (MRA) and Wavelets [2]

A set of subspaces is said to be MRA of if it possesses the following properties:

where denotes the set of integers. Properties (34)–(36) state that is a nested sequence of subspaces that effectively covers . That is, every square integrable function can be approximated as closely as desired by a function that belongs to at least one of the subspaces . A function is called a scaling function if it generates the nested sequence of subspaces and satisfies the dilation equation; namely,

with and being any rational number.

For each scale , since , there exists a unique orthogonal complementary subspace of in . This subspace is called wavelet subspace and is generated by , where is called the wavelet. From the above discussion, these results follow easily:

Some of the important properties relevant to the present analysis are given below [2, 19].

(1) Vanishing Moment. A wavelet is said to have a vanishing moment of order if

All wavelets must satisfy the previously mentioned condition for .

(2) Semiorthogonality. The wavelets form a semiorthogonal basis if

3.2.2. Method of Moments for the Solution of Fredholm Integral Equation

In this section, we solve the integral equation of form (7) in interval by using linear B-spline wavelets [2]. The unknown function in (7) can be expanded in terms of the scaling and wavelet functions as follows:

By substituting this expression into (7) and employing the Galerkin method, the following set of linear system of order is generated. The scaling and wavelet functions are used as testing and weighting functions:

where

and the subscripts , and assume values as given below:

In fact, the entries with significant magnitude are in the and submatrices which are of order and , respectively.

3.3. Variational Iteration Method [3–5]

In this section, Fredholm integral equation of second kind given in (7) has been considered for solving (7) by variational iteration method. First, we have to take the partial derivative of (7) with respect to yielding

We apply variation iteration method for (46). According to this method, correction functional can be defined as

where is a general Lagrange multiplier which can be identified optimally by the variational theory, the subscript denotes the th order approximation, and is considered as a restricted variation; that is, . The successive approximations , for the solution can be readily obtained after determining the Lagrange multiplier and selecting an appropriate initial function . Consequently the approximate solution may be obtained by using

To make the above correction functional stationary, we have

Under stationary condition,

implies the following Euler Lagrange equation:

with the following natural boundary condition:

Solving (51), along with boundary condition (52), we get the general Lagrange multiplier

Substituting the identified Lagrange multiplier into (47) results in the following iterative scheme:

By starting with initial approximate function (say), we can determine the approximate solution of (7).

4. Numerical Methods for System of Linear Fredholm Integral Equations of Second Kind

Consider the system of linear Fredholm integral equations of second kind of the following form:

where and are known functions and are the unknown functions for .

4.1. Application of Haar Wavelet Method [9]

In this section, an efficient algorithm for solving Fredholm integral equations with Haar wavelets is analyzed. The present algorithm takes the following essential strategy. The Haar wavelet is first used to decompose integral equations into algebraic systems of linear equations, which are then solved by collocation methods.

4.1.1. Haar Wavelets

The compact set of scale functions is chosen as

The mother wavelet function is defined as

The family of wavelet functions generated by translation and dilation of are given by

where , , .

Mutual orthogonalities of all Haar wavelets can be expressed as

4.1.2. Function Approximation

An arbitrary function can be expanded into the following Haar series:

where the coefficients are given by

In particular, .

The previously mentioned expression in (60) can be approximately represented with finite terms as follows:

where the coefficient vector and the Haar function vector are, respectively, defined as

The Haar expansion for function of order is defined as follows:

where , , , .

From (62) and (64), we obtain

where

4.1.3. Operational Matrices of Integration

We define

where , .

Then, for , the corresponding matrix can be represented as

The integration of the Haar function vector is

where is the operational matrix of order , and

By recursion of the above formula, we obtain

Therefore, we get

where and is a positive integer.

The inner product of two Haar functions can be represented as

where

4.1.4. Haar Wavelet Solution for Fredholm Integral Equations System [9]

Consider the following Fredholm integral equations system defined in (55):

The Haar series of and , ; are, respectively, expanded as

Substituting (76) into (75), we get

From (77) and (73), we get

Interpolating collocation points, that is, , in the interval leads to the following algebraic system of equations:

Hence, , can be computed by solving the above algebraic system of equations and consequently the solutions , .

4.2. Taylor Series Expansion Method

In this section, we present Taylor series expansion method for solving Fredholm integral equations system of second kind [7]. This method reduces the system of integral equations to a linear system of ordinary differential equation. After including boundary conditions, this system reduces to a system of equations that can be solved easily by any usual methods.

Consider the second kind Fredholm integral equations system defined in (55) as follows:

A Taylor series expansion can be made for the solution of in the integral equation (80):

where denotes the error between and its Taylor series expansion in (81).

If we use the first term of Taylor series expansion and neglect the term containing , that is, , then, substituting (81) for into the integral in (80), we have

Equation (83) becomes a linear system of ordinary differential equations that we have to solve. For solving the linear system of ordinary differential equations (83), we require an appropriate number of boundary conditions.

In order to construct boundary conditions, we first differentiate times both sides of (80) with respect to ; that is,

where , .

Applying the mean value theorem for integral in (84), we have

Now (83) combined with (85) becomes a linear system of algebraic equations that can be solved analytically or numerically.

4.3. Block-Pulse Functions for the Solution of Fredholm Integral Equation

In this section, Block-Pulse functions (BPF) have been utilized for the solution of system of Fredholm integral equations [6].

An -set of BPF is defined as follows:

with , and .

4.3.1. Properties of BPF

(1) Disjointness. One has

, . This property is obtained from definition of BPF.

(2) Orthogonality. One has

, . This property is obtained from the disjointness property.

(3) Completeness. For every is complete; if then almost everywhere. Because of completeness of , we have

for every real bounded function which is square integrable in the interval and .

4.3.2. Function Approximation

The orthogonality property of BPF is the basis of expanding functions into their Block-Pulse series. For every ,

where is the coefficient of Block-Pulse function, with respect to th Block-Pulse function .

The criterion of this approximation is that mean square error between and its expansion is minimum

so that we can evaluate Block-Pulse coefficients.

In the matrix form, we obtain the following from (90) as follow:

Now let be two-variable function defined on and ; then can be expanded to BPF as

where and are and dimensional Block-Pulse function vectors and is a Block-Pulse coefficient matrix.

There are two different cases of multiplication of two BPF. The first case is

It is obtained from disjointness property of BPF. It is a diagonal matrix with Block-Pulse functions.

The second case is

because .

Operational Matrix of Integration. BPF integration property can be expressed by an operational equation as

where

A general formula for can be written as

By using this matrix, we can express the integral of a function into its Block-Pulse series

4.3.3. Solution for Linear Integral Equations System

Consider the integral equations system from (55) as follows:

Block-Pulse coefficients of , in the interval can be determined from the known functions , and the kernels , . Usually, we consider to facilitie the use of Block-Pulse functions. In case , we set , where .

We approximate , , by its BPF as follows:

where , , and are defined in Section 4.3.2, and substituting (102) into (101), we have

since

From (104), we get

Set ; then we have from (106)

which is a linear system

After solving the above system we can find , and hence obtain the solutions , .

5. Numerical Methods for Nonlinear Fredholm-Hammerstein Integral Equation

We consider the second kind nonlinear Fredholm integral equation of the following form:

where is the kernel of the integral equation, and are known functions, and is the unknown function that is to be determined.

5.1. B-Spline Wavelet Method

In this section, nonlinear Fredholm integral equation of second kind of the form given in (109) has been solved by using B-spline wavelets [11].

B-spline scaling and wavelet functions in the interval and function approximation have been defined in Sections 3.1.1 and 3.1.2, respectively.

First, we assume that

Now, from (16), we can approximate the functions and as

where and are column vectors similar to defined in (17).

Again, by using dual of the wavelet functions, we can approximate the functions and as follows:

where

From (110)–(112), we get

Applying (110)–(114) in (109), we get

Multiplying (115) by both sides from the right and integrating both sides with respect to from 0 to 1, we have

where is a square matrix given by

Equation (116) gives a system of algebraic equations with unknowns for and vectors given in (111).

To find the solution in (111), we first utilize the following equation:

with the collocation points , where .

Equation (118) gives a system of algebraic equations with unknowns, for and vectors given in (111).

Combining (116) and (118), we have a total of system of algebraic equations with unknowns for and . Solving those equations for the unknown coefficients in the vectors and , we can obtain the solution .

5.2. Quadrature Method Applied to Fredholm Integral Equation

In this section, Quadrature method has been applied to solve nonlinear Fredholm-Hammerstein integral equation [10].

The quadrature methods like Simpson rule and modified trapezoid method are applied for solving a definite integral as follows.

5.2.1. Simpson’s Rule

One has

5.2.2. Modified Trapezoid Rule

One has

Consider the nonlinear Fredholm integral equation of second kind defined in (109) as follows:

For solving (121), we approximate the right-hand integral of (121) with Simpson’s rule and modified trapezoid rule; then we get the following.

5.2.3. Simpson’s Rule

One has

Hence, for and in (122), we have

Equation (123) is a nonlinear system of equations and, by solving (123), we obtain the unknowns for .

5.2.4. Modified Trapezoid Rule

One has

where .

For and in (124), we have

for .

This is a system of equations and unknowns. By taking derivative from (121) and setting , we obtain

If is a solution of (121), then it is also solution of (126). By using trapezoid rule for (126) and replacing , we get

for . In case of from system (127), we obtain two equations.

Now (127) combined with (125) generates the nonlinear system of equations as follows:

By solving this system with nonlinear equations and unknowns, we can obtain the solution of (109).

5.3. Wavelet Galerkin Method

In this section, the continuous Legendre wavelets [12], constructed on the interval , are applied to solve the nonlinear Fredholm integral equation of the second kind. The nonlinear part of the integral equation is approximated by Legendre wavelets, and the nonlinear integral equation is reduced to a system of nonlinear equations.

We have the following family of continuous wavelets with dilation parameter and the translation parameter

Legendre wavelets have four arguments; , , , is the order for Legendre polynomials and is the normalized time.

Legendre wavelets are defined on [0, 1) by

where are the well-known Legendre polynomials of order m, which are orthogonal with respect to the weight function and satisfy the following recursive formula:

The set of Legendre wavelets are an orthonormal set.

5.3.1. Function Approximation

A function can be expanded as

where

If the infinite series in (132) is truncated, then (132) can be written as

where and are matrices given by

Similarly, a function can be approximated as

where is matrix, with

Also, the integer power of a function can be approximated as

where is a column vector, whose elements are nonlinear combinations of the elements of the vector . is called the operational vector of the th power of the function .

5.3.2. The Operational Matrices

The integration of the vector defined in (136) can be obtained as

where is the operational matrix for integration and is given in [23] as

In (141), and are matrices given in [23] as

The integration of the product of two Legendre wavelets vector functions is obtained as

where is an identity matrix.

The product of two Legendre wavelet vector functions is defined as

where is a vector given in (135) and is matrix, which is called the product operation of Legendre wavelet vector functions [23, 24].

5.3.3. Solution of Fredholm Integral Equation of Second Kind

Consider the nonlinear Fredholm-Hammerstein integral equation of second kind of the form

where , , is an unknown function, and is a positive integer.

We can approximate the following functions as

Substituting (146) into (145), we have

Equation (147) is a system of algebraic equations. Solving (147), we can obtain the solution .

5.4. Homotopy Perturbation Method

Consider the following nonlinear Fredholm integral equation of second kind of the form

For solving (148) by Homotopy perturbation method (HPM) [14–16], we consider (148) as

As a possible remedy, we can define by

where is an integral operator with known solution .

We may choose a convex homotopy by

and continuously trace an implicitly defined curve from a starting point to a solution function . The embedding parameter monotonically increases from zero to unit as the trivial problem . The embedding parameter can be considered as an expanding parameter. The HPM uses the homotopy parameter as an expanding parameter; that is,

When , (152a) corresponding to (151) become the approximate solution of (149) as follows:

The series in (152b) converges in most cases, and the rate of convergence depends on [14].

Consider

The nonlinear term can be expressed in He polynomials [25] as

where

Substituting (152a), (153), and (154) into (151), we have

Equating the terms with identical power of in (156), we have

and in general form we have

Hence, we can obtain the approximate solution of aforesaid equation (148) from (152b).

5.5. Adomian Decomposition Method

Adomian decomposition method (ADM) [16–18] has been applied to a wide class of functional equations. This method gives the solution as an infinite series usually converging to an accurate solution. Let us consider the nonlinear Fredholm integral equation of second kind as follows:

where and are the linear and nonlinear terms, respectively.

The Adomian decomposition method (ADM) consists of representing as a series

In the view of ADM, the nonlinear term can be represented as

Now substituting (160) and (161) into (159), we have

and, then, ADM uses the recursive relations

where is so-called Adomian polynomial.

Therefore, we obtain the -terms approximate solution as

with

6. Conclusion and Discussion

In this work, we have examined many numerical methods to solve Fredholm integral equations. Using these methods except variational iteration method, the Fredholm integral equations have been reduced to a system of algebraic equations and this system can be easily solved by any usual methods. In this work, we have applied compactly supported semiorthogonal B-spline wavelets along with their dual wavelets for solving both linear and nonlinear Fredholm integral equations of second kind. The problem has been reduced to solve a system of algebraic equations. In order to increase the accuracy of the approximate solution, it is necessary to apply higher-order B-spline wavelet method. The method of moments based on compactly supported semiorthogonal B-spline wavelets via Galerkin method has been used to solve Fredholm integral equation of second kind. This method determines a strong reduction in the computation time and memory requirement in inverting the matrix. Variational iteration method has been successfully applied to find the approximate solution of Fredholm integral equation of both linear and nonlinear types. Taylor series expansion method reduces the system of integral equations to a linear system of ordinary differential equation. After including the required boundary conditions, this system reduces to a system of algebraic equations that can be solved easily. Block-Pulse functions and Haar wavelet method can be applied to the system of Fredholm integral equations by reducing into a system of algebraic equations. These methods give more accuracy if we increase their order. Quadrature method can be applied to solve the nonlinear Fredholm-Hammerstein integral equation of second kind by reducing it to a system of algebraic equations. Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) can be also applied to approximate the solution of nonlinear Fredholm integral equation of second kind. The solutions obtained by HPM and ADM are applicable for not only weakly nonlinear equations, but also strong ones. The approximate solutions by these aforesaid methods highly agree with exact solutions.

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