Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.


INTRODUCTION
Finding the analytical solutions of functional equations has been devoted of attention of mathematicians' interest in recent years.Several methods are proposed to achieve this purpose, such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].Mathematical modeling for many problems in different fields, such as engineering, chemistry, physics and biology, leads to integral equations or system of integral equations.Several methods have been proposed to solve these problems.For example, Variational iteration method [19], differential transform method [20], Nystrom method [21], Haar functions method [22], Homotopy perturbation method [23], Chebyshev wavelet method [24] and many others.Between of present methods, spectral methods have been used to solve different functional equations, because of their high accuracy and easy applying.Specific types of spectral methods that more applicable and widely used, are the Galerkin, collocation, and tau methods [25][26][27][28][29]. Saadatmandi and Dehghan introduced shifted Legendre operational matrix for fractional differential equations [30], Doha derived a new explicit formula for shifted Chebyshev polynomials for fractional differential equations [31], Bhrawy used a quadrature shifted Legendre tau method for fractional differential equations [32].Recently, Doha introduced shifted Chebyshev operational matrix and applied it with spectral methods for solving problems to initial and boundary conditions [33].
The importance of Sturm-Liouville problems for spectral methods lies in the fact that the spectral approximation of the solution of a functional equation is usually regarded as a finite expansion of eigenfunctions of a suitable Sturm-Liouville problem.
The Jacobi polynomials ( , ) ( )( 0, , 1) i P x i play important roles in mathematical analysis and its applications [34].It is proven that Jacobi polynomials are precisely the only polynomials arising as eigenfunctions of a singular Sturm-Liouville problem [35 -36].This class of polynomials comprises all the polynomial solution to singular Sturm-Liouville problems on [ 1,1] .Chebyshev, Legendre, and ultraspherical polynomials are particular cases of the Jacobi polynomials.
In this paper, the shifted Jacobi operational matrices of integration and product is introduced, which is based on Jacobi collocation method for solving numerically the systems of the linear and nonlinear Fredholm and Volterra integral equations on the interval [0,1], to find the approximate solution () The each of equation of the systems resulted are collocated at (  1) N nodes of the shifted Jacobi-Gauss interpolation on(0,1) .These equations generate (  1) nN linear or nonlinear algebraic equations.The nonlinear systems resulted can be solved using Newton iterative method.The remainder of this paper is organized as follows: The Jacobi polynomials and their integral and product operational matrices to integral equations are obtained in Section 2. Section 3 is devoted to applying the Jacobi operational matrices for solving system of integral equations.In Section 4, the proposed method is applied to several examples.A conclusion is presented in Section 5.

Properties of shifted Jacobi polynomials
The Jacobi polynomials, associated with the real parameters ( ,  1) are a sequence of polynomials , and .
These polynomials can be generated with the following recurrence formula;

P t t
In order to use these polynomials on the interval [0,1], shifted Jacobi polynomials are defined by introducing the change of variable 21 tx.Let the shifted Jacobi polynomials i Px be denoted by Rx can be generated from following formula; ) where,

, ( ).
Si Ux These orthogonal polynomials are related to the shifted Jacobi polynomials by the following relations.
The analytic form of the shifted Jacobi polynomials, Some properties of the shifted Jacobi polynomials are as follows, The orthogonality condition of shifted Jacobi polynomials is where ( , ) () Wx , shifted weighted function, is as follows, .
p can be obtained as, Now, using properties (1) and (3) in above, the lemma can be proved.Lemma 2. For 0 m , one has , B s t is the Beta function and is defined as .

The approximation of functions
Let (0,1) , and for r ( is the set of all non-negative integers), the weighted Sobolov space is defined in the usual way and is denoted inner product, semi-norm and norm by , , .
, ( ),..., ( ) as the below formula, , where the coefficients j c are given by By noting in practice, only the first ( 1) N terms shifted polynomials are considered, then one has where Since ( , , ) N P is a finite dimensional vector space, () ux has a unique best approximation from ( , , ) N

P
, say N N WW y u x u x u x y P In [37] is shown that for any and 0 r , a positive constant C independent of any function, N , and exist that .
times continuously differentiable.The following Theorem can present an upper bound for estimating the error.

The Jacobi integral operational matrix
In this subsection, Jacobi operational matrix of the integration is derived.Let 0 ( ) ( ), x t dt P x (9) where matrix P is called the Jacobi operational matrix of the integration.Theorem 3. Let P is ( 1) ( 1) NN operational matrix of integral.Then the elements of this matrix are obtained as Proof.Using Eq. ( 9) and orthogonality property of Jacobi polynomials one has, , ( , ) () Rx by using Lemma 1 can be obtained as ..., .
x N , then an error bound of integral operator of vector can be expressed by .

The product operational matrix
The following property of the product of two Jacobi function vector will also be applied to solve the Volterra integral equations. where

NN
product operational matrix and it`s elements are determined in terms of the vector Y `s elements.Using Eq.( 11) and by the orthogonality property of Jacobi polynomials the elements ij Y can be calculated as follows,

APPLICATIONS OF THE OPERATIONAL MATRICES OF INTEGRATION AND PRODUCT
In this section, the presented operational matrices are applied to solve the system of linear and nonlinear Fredholm and Volterra integral equations.

The system of Fredholm integral equations
A system of Fredholm integral equations can be presented as follows; ..., , where .
x With substituting approximations (13) in system (12) one has where D is the following (  1) ( 1) NN known matrix, The system ( 13) have ( 1) nN unknown coefficients i j c .For collocating, ( 1) N roots of Jacobi polynomials () N Rx are applied and the equations are collocated at them.Unknown coefficients are determined with solving the resulted system of linear or nonlinear algebraic equations.
In the same way, the system of following equations is resulted.
By using the first (

ILLUSTRATIVE EXAMPLES
In this section, the presented method is applied to solve some examples.Comparison between the results of present method with the corresponding analytic solutions is given.For this purpose, the maximum of absolute error is computed.
Example 1.The following system of linear Volterra integral equations of the second kind is considered,  (18) The exact solutions are 1 ( )  , solutions and kernels are approximated as: TT tx t x x K t tx t x K t The system (18) by using above equations is rewritten as, ) Rx and collocating the system (19), reduces the problem to solve a system of algebraic equations.Unknown coefficients are obtained for some values of parameters and .Maximum absolute error for 4 N and different values of and has been listed in Table 1.Table 1 shows that a good approximation can be achieved for the exact solutions by using a few terms of shifted Jacobi polynomials for various values of parameters and .
Example 2. Consider the following system of linear Fredholm integral equation of second kind, , The exact solutions are 1 () x u x e and 2 1 () , solutions and kernels are approximated as: The system (20) by using above equations is rewritten as, Now using the roots of () Rx and collocating the system (21), reduces the problem to solve a system of linear algebraic equations.Solving the system (21), the unknown coefficients will be obtained.The system (22) by using above equations is rewritten as, Now using the roots of () Rx and collocating the system (23), reduces the problem to solve a system of linear algebraic equations.Solving the system (23), the unknown coefficients will be obtained.Table 3  , solutions and kernels are approximated as: The system (24) by using above equations is rewritten as, () Rx and collocating the each equation of system (25), reduces the problem to solve a system of nonlinear algebraic equations.Solving the system (25) by Newton iterative method, the unknown coefficients will be obtained.
The exact solutions are 1 ( ) cos( ) u x x and 1 ( ) sin( ) u x x .Solutions and kernels are approximated as: The system (26) by using above equations is rewritten as, are listed in Table 5.Also, using the roots of () Rx and collocating the each equation of system (27), reduces the problem to solve a system of nonlinear algebraic equations.Solving the system (27) by Newton iterative method, the unknown coefficients will be obtained.

CONCLUTION
In this paper, the shifted Jacobi collocation method was employed to solve a class of systems of Fredholm and Volterra integral equations of first and second kinds.First, a general formulation for the Jacobi operational matrix of integral has been derived.This matrix is used to approximate numerical solution of system of linear and nonlinear Volterra integral equations.Proposed approach was based on the shifted Jacobi collocation method.The solutions obtained using the proposed method shows that this method is a powerful mathematical tool for solving the integral equations.Proving the convergence of the method, consistency and stability are ensured automatically.Moreover, only a small number of shifted Jacobi polynomials are needed to obtain a satisfactory result.

3 C, C 2 C and 3 C
are the operational matrices of product corresponding to unknown vectors 1 .Now using the roots of , respectively.Now using the roots of ( , )

3 E
are the operational matrices of product corresponding with the vectors 21 , , respectively.The maximum absolute errors for 0 and 7,10,15 N , )

,
Rx and Now the following theorem can present an upper bound for estimating the error of integral operator.The error vector E is defined as,

Table 1 .
Maximum absolute error for

Table 2 .
Maximum absolute error for

Table 3 .
Maximum absolute error for

Table 4 .
Maximum absolute error for

Table 5 .
Table 6 displays the maximum absolute errors for various and with 15 N. Maximum absolute error for

Table 6 .
Maximum absolute error for