Numerical Methods for Solving Fredholm Integral Equations of Second Kind

and Applied Analysis 3 are called cardinal B-spline functions of order m ≥ 2 for the knot sequence X = {xi} n+m−1 i=−m+1 and SuppBm,j,X(x) = [xj, xj+m] ∩ [a, b]. By considering the interval [a, b] = [0, 1], at any level j ∈ Ζ , the discretization step is 2, and this generates n = 2 number of segments in [0, 1] with knot sequence X (j) = {{ {{ { x (j) −m+1 = ⋅ ⋅ ⋅ = x (j) 0 = 0, x (j) k = k 2 , k = 1, . . . , n − 1, x (j) n = ⋅ ⋅ ⋅ = x (j) n+m−1 = 1. (10) Let j0 be the level forwhich 2 j0 ≥ 2m−1; for each level, j ≥ j0, the scaling functions of order m can be defined as follows in [2]: φm,j,i (x) = {{ {{ { m,j0,i (2 0x) i = −m + 1, . . . , −1, m,j0,2 −m−i (1 − 2 0x) i = 2 j − m + 1, . . . , 2 j − 1, m,j0,0 (2 0x − 2 −j0 i) i = 0, . . . , 2 j − m. (11) And the two scale relations for the m-order semiorthogonal compactly supported B-wavelet functions are defined as follows: ψm,j,i−m = 2i+2m−2 ∑ k=i qi,kBm,j,k−m, i = 1, . . . , m − 1, ψm,j,i−m = 2i+2m−2 ∑ k=2i−m qi,kBm,j,k−m, i = m, . . . , n − m + 1, ψm,j,i−m = n+i+m−1 ∑ k=2i−m qi,kBm,j,k−m, i = n − m + 2, . . . , n, (12) where qi,k = qk−2i. Hence, there are 2(m − 1) boundary wavelets and (n − 2m+ 2) inner wavelets in the bounded interval [a, b]. Finally, by considering the level jwith j ≥ j0, the B-wavelet functions in [0, 1] can be expressed as follows: ψm,j,i (x) = {{ {{ { m,j0,i (2 0x) i = −m + 1, . . . , −1, ψm,2j−2m+1−i,i (1 − 2 0x) i = 2 j −2m+2, . . . , 2 j −m, m,j0,0 (2 0x − 2 −j0 i) i = 0, . . . , 2 j − 2m + 1. (13) The scaling functions φm,j,i(x) occupy m segments and the wavelet functions ψm,j,i(x) occupy 2m − 1 segments. When the semiorthogonal wavelets are constructed from B-spline of order m, the lowest octave level j = j0 is determined in [19, 20] by 2 j0 ≥ 2m − 1, (14) so as to have a minimum of one complete wavelet on the interval [0, 1]. 3.1.2. Function Approximation. A function f(x) defined over [0, 1]may be approximated by B-spline wavelets as [21, 22] f (x) = 2 0−1 ∑ k=1−m j0 ,k j0 ,k (x) + ∞ ∑ j=j0 2 j −m ∑ k=1−m dj,kψj,k (x) . (15) If the infinite series in (15) is truncated atM, then (15) can be written as [2] f (x) ≅ 2 0−1 ∑ k=1−m j0 ,k j0 ,k (x) + M ∑ j=j0 2 j −m ∑ k=1−m dj,kψj,k (x) , (16) where φ2,k and ψj,k are scaling and wavelets functions, respectively, andC andΨ are (2 +m−1)×1 vectors given by C = j0,1−m , . . . , j0 ,2 j0−1, j0 ,1−m , . . . , j0 ,2 j0−m, . . . , dM,1−m, . . . , dM,2M−m] T , (17) Ψ = j0 ,1−m , . . . , j0 ,2 j0−1, j0,1−m , . . . , j0,2 j0−m, . . . , ψM,1−m, . . . , ψM,2M−m] T , (18) with j0 ,k = ∫ 1 0 f (x) j0 ,k (x) dx, k = 1 − m, . . . , 2 j0 − 1, dj,k = ∫ 1 0 f (x) ?̃?j,k (x) dx, j = j0, . . . ,M, k = 1 − m, . . . , 2 M − m, (19) where j0 ,k(x) and ?̃?j,k(x) are dual functions of j0 ,k and ψj,k, respectively.These can be obtained by linear combinations of j0 ,k , k = 1 − m, . . . , 20 − 1, and ψj,k, j = j0, . . . ,M, k = 1 − m, . . . , 2 M − m, as follows. Let Φ = j0 ,1−m , . . . , j0 ,2 j0−1] T , (20) Ψ = j0 ,1−m , . . . , j0 ,2 j0−m, . . . , ψM,1−m, . . . , ψM,2M−m] T . (21) Using (11), (20), (12)-(13), and (21), we get


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][4][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][7][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][15][16] and Adomian decomposition method (ADM) [16][17][18] have been applied to solve nonlinear Fredholm integral equation of second kind.

B-Spline Scaling and Wavelet Functions on the Interval
[0, 1].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  2 ().
Using this basis, an arbitrary function in  2 () can be expressed as the wavelet series.For the finite interval [0, 1], 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 By considering the interval [, ] = [0, 1], at any level  ∈ Ζ + , the discretization step is 2 − , and this generates  = 2  number of segments in [0, 1] with knot sequence Let  0 be the level for which 2  0 ≥ 2 −1; for each level,  ≥  0 , 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  , =  −2 .Hence, there are 2( − 1) boundary wavelets and ( − 2 + 2) inner wavelets in the bounded interval [, ].Finally, by considering the level  with  ≥  0 , the B-wavelet functions in [0, 1] can be expressed as follows: The scaling functions  ,, () occupy  segments and the wavelet functions  ,, () occupy 2 − 1 segments.When the semiorthogonal wavelets are constructed from B-spline of order , the lowest octave level  =  0 is determined in [19,20] by so as to have a minimum of one complete wavelet on the interval [0, 1].

Multiresolution Analysis (MRA) and Wavelets
with   ∈  2 and  being any rational number.
For each scale , since   ⊂  +1 , there exists a unique orthogonal complementary subspace   of   in  +1 .This subspace   is called wavelet subspace and is generated by  , = (2   − ), where  ∈  2 is called the wavelet.From the above discussion, these results follow easily: (39) Some of the important properties relevant to the present analysis are given below [2,19].
All wavelets must satisfy the previously mentioned condition for  = 0.
( In this section, we solve the integral equation of form (7) in interval [0, 1] 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 (2  + 1) 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 (2  0 + 1) and (2 +1 + 1), respectively.[3][4][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

Variational Iteration Method
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 ỹ is considered as a restricted variation; that is,  ỹ = 0.The successive approximations   (),  ≥ 1 for the solution () can be readily obtained after determining the Lagrange multiplier and selecting an appropriate initial function  0 ().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  0 () = () (say), we can determine the approximate solution () of ( 7).

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 ,  = 1, 2, . . ., . [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.

Application of Haar Wavelet Method
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 ℎ 1 () are given by where Mutual orthogonalities of all Haar wavelets can be expressed as where the coefficients   are given by In particular,  0 = ∫ 1 0 ().The previously mentioned expression in (60) can be approximately represented with finite terms as follows: where the coefficient vector   () and the Haar function vector ℎ () () 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  = 4, the corresponding matrix can be represented as ) , . . ., ℎ (4) ( The integration of the Haar function vector ℎ () () is where  () is the operational matrix of order , and By recursion of the above formula, we obtain Therefore, we get where  = 2  and  is a positive integer.The inner product of two Haar functions can be represented as where ) . (74)

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, 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  () , (, ) =  ()  , (, )/ () ,  = 1, 2, . . ., .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.

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].
Abstract and Applied Analysis 9

Function Approximation.
The orthogonality property of BPF is the basis of expanding functions into their Block-Pulse series.For every () ∈  2 (), where   is the coefficient of Block-Pulse function, with respect to th Block-Pulse function Φ  ().
The criterion of this approximation is that mean square error between () and its expansion is minimum so that we can evaluate Block-Pulse coefficients.
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 Operational Matrix of Integration.BPF integration property can be expressed by an operational equation as where A general formula for  × can be written as ) .

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.

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 [0, 1] and function approximation have been defined in Sections 3.1.1and 3.1.2,respectively.

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.

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.

Wavelet Galerkin Method.
In this section, the continuous Legendre wavelets [12], constructed on the interval [0, 1], 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.
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 () = 1 and satisfy the following recursive formula: The set of Legendre wavelets are an orthonormal set.
(138) 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 ().

The Operational Matrices.
The integration of the vector Ψ() defined in (136) can be obtained as where  is the (2 −1  × 2 −1 ) 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 C is (2 −1  × 2 −1 ) matrix, which is called the product operation of Legendre wavelet vector functions [23,24].

Solution of Fredholm Integral Equation of Second Kind.
Consider the nonlinear Fredholm-Hammerstein integral equation of second kind of the form where  ∈  2 [0, 1],  ∈  2 ([0, 1] × [0, 1]),  is an unknown function, and  is a positive integer.We can approximate the following functions as Equation ( 147) is a system of algebraic equations.Solving (147), we can obtain the solution () ≈   Ψ().
where () is an integral operator with known solution  0 .We may choose a convex homotopy by  (, ) = (1 − )  () +  () = 0 (151) and continuously trace an implicitly defined curve from a starting point ( 0 , 0) to a solution function (, 1).The embedding parameter  monotonically increases from zero to unit as the trivial problem () = 0.The embedding parameter  ∈ (0, 1] can be considered as an expanding parameter.The HPM uses the homotopy parameter  as an expanding parameter; that is, When  → 1, (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 Hence, we can obtain the approximate solution of aforesaid equation ( 148) from (152b).

Adomian Decomposition Method.
Adomian decomposition method (ADM) [16][17][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: (160) In the view of ADM, the nonlinear term  can be represented as where   = 1 ! (      ( where   is so-called Adomian polynomial.Therefore, we obtain the -terms approximate solution as with

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 Bspline 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.