Improved Jacobi matrix method for the numerical solution of Fredholm integro-differential-difference equations

This study is aimed to develop a new matrix method, which is used an alternative numerical method to the other method for the high-order linear Fredholm inte-gro-differential-difference equation with variable coefﬁcients. This matrix method is based on orthogonal Jacobi polynomials and using collocation points. The improved Jacobi polynomial solution is obtained by summing up the basic Jacobi polynomial solution and the error estimation function. By comparing the results, it is shown that the improved Jacobi polynomial solution gives better results than the direct Jacobi polynomial solution, and also, than some other known methods. The advantage of this method is that Jacobi polynomials comprise all of the Legendre, Chebyshev, and Gegenbauer polynomials and, therefore, is the comprehensive polynomial solution technique


Introduction Orthogonal Jacobi polynomials
The systems of polynomials remain a very active research area in mathematics, physics, engineering and other applied sciences; and the orthogonal polynomials, among others, are definitely the most thoroughly studied and widely applied systems [1][2][3]. The three of these systems, namely, Hermite, Laguerre, and Jacobi, are called collectively the classical orthogonal polynomials [4]. There is excessive literature on these polynomials, and the most comprehensive single account of the classical polynomials is found in the classical treatise of Szegö [5].
Jacobi polynomials are the common set of orthogonal polynomials defined by the formula [4] Here, a and b are parameters that, for integrability purposes, are restricted to a [ À 1; b [ À 1. However, many of the identities and other formal properties of these polynomials remain valid under the less restrictive condition that neither a is b a negative integer. Among the many special cases, the following is the most important [4] (a) The Legendre polynomials The Jacobi polynomials P a;b ð Þ n x ð Þ are defined [6,7] with respect to the weight function x a;b x ð Þ ¼ 1 À Mustafa Bahşı mustafa.bahsi@cbu.edu.tr ða [ À 1; b [ À 1Þ on À1; 1 ð Þ. It is proved that the Jacobi polynomials satisfy the following relation [7]: where B a;b;n ð Þ n ¼ 2 Àk n þ a þ b þ k k n þ a n À k ; k ¼ 0; 1; 2; . . .; n These polynomials play role in rotation matrices [8], in the trigonometric Reson-Morse potential [9], and the cases of a few exact solutions in quantum mechanics [10,11].
In very recent years, several researchers developed new numerical algorithms for some problems using Jacobi polynomials. Eslahchi et al. [12] gave a numerical solution for some nonlinear ordinary differential equations using the spectral method. Bojdi et al. [13] proposed a Jacobi matrix method for differential-difference equations with variable coefficients. Kazem [14] used the Tau method for solving fractional-order differential equations by means of Jacobi polynomials.
In this study, we generate a procedure to find a Jacobi polynomial solution for the nth order linear FIDDE with variable coefficients where P i x ð Þ, Q j x ð Þ, K x; t ð Þ, and g x ð Þ are known functions and a ki , b ki , c ki , and l k are appropriate constants, while y x ð Þ is the unknown function. Note that a g b is a given point in the spatial domain of problem.
The main aim of our study, using orthogonal Jacobi polynomials, is to provide an approximate solution for the problem (4,5), which is usually hard to find analytical solutions.
We assume a solution expressed as the truncated series of orthogonal Jacobi polynomials defined by where P a;b ð Þ n x ð Þ, n ¼ 0; 1; . . .; N denote the orthogonal Jacobi polynomials defined by (2,3); N is chosen N ! n and a n ; n ¼ 0; 1; . . .; N are unknown coefficients to be determined. Note that a and b are arbitrary parameters,

Fundamental matrix relations
We can transform the orthogonal Jacobi polynomials P a;b ð Þ n x ð Þ from algebraic form into matrix form as follow: and We assume the solution y x ð Þ, which is defined by the truncated orthogonal Jacobi series (6) in matrix form as follow By substituting the matrix form of Jacobi polynomials (7) to (11) into (4,5), we can obtain the fundamental matrix equation of approximate solution of unknown function as Matrix representation of differential-difference part of problem Differential-difference part of problem is P n i¼0 P i x ð Þy ðiÞ x ð Þ þ P m j¼0 Q j x ð Þy ðjÞ x À s ð Þ:First, to explain the relation between the matrix form of the unknown function and the matrix form of its derivative y ðiÞ x ð Þ, we introduce the relation between X x ð Þ and its derivatives X i ð Þ x ð Þ can be expressed as Then, using (13) and (14), we may write Similarly, the relation between the matrix form of unknown function and matrix form of its delay forms' derivatives y ðjÞ x À s ð Þ can be expressed as where Thus, it is seen that Using (16) and (19), the matrix form of differential-difference part of Eq. (4) becomes Matrix representation of integral part of problem ð Þ the kernel function of the Fredholm integral part of main problem is. This function can be written using the truncated Taylor Series [30] and the truncated orthogonal Jacobi series, respectively, as and is the Taylor coefficient and k J mn is the Jacobi coefficient. The expressions (21) and (22) can be written using matrix forms of the Jacobi polynomials, respectively, as The following relation can be obtained from Eqs. (7), (23), and (24), . . .
By substituting the Eqs. (19) and (25) into such that Finally, substituting the form (7) into expression (26) yields the matrix relation

Matrix representation of conditions
In this section, we write to the matrix form of mixed conditions of the problem given Eq. (5), using the matrix relation (16), as

Method of solution
We substitute obtained matrix relations in the previous subsections given in Eqs. (20) and (27) into fundamental problem to build the fundamental matrix equation of the problem. For this purpose, we can define collocation points as follow: As can be observed, standard collocation points dividing the domain interval ½a; b of the problem into N equal parts are employed. Accordingly, we obtain the system of matrix equations The fundamental matrix equation becomes where Equation (29), which is matrix representation of the Eq. (4), corresponds to a system of N þ 1 algebraic equations. This system indicates N þ 1 unknown coefficients, such that a 0 ; a 1 ; a 2 ; . . .; a N . Briefly, if we define Similarly, from (28), the matrix form of mixed conditions can be obtained briefly as such that Consequently, to find the Jacobi polynomial solution of Eq. (4) under the mixed conditions (5), we replace the row matrix (31) by last n rows of the augmented matrix (30), which yields the new matrix equation form written as follow If rankW ¼ rankW;G Â Ã ¼ N þ 1, then we can find the matrix of unknown coefficient of Jacobi series via A ¼W À Á À1G . Note that the matrix A (thereby, the coefficients a 0 ; a 1 ; a 2 ; . . .; a N ) is uniquely determined [22]. Equation (4) has also a unique solution under the conditions (5). Thus, we get the Jacobi polynomial solution for arbitrary parameters a and b:

Error analysis
In this part of study, it is given to a useful error estimation procedure for orthogonal Jacobi polynomial solution of the problem. Also, this procedure is used to obtain the improved solution of the problem (4, 5) according to the direct Jacobi polynomial solution. For this purpose, we use the residual correction technique [31,32] and error estimation by the known Tau method [33,34].
Recently, Yüzbaşı and Sezer [35] solved a class of the Lane-Emden equations using the improved BCM with residual error function. Yüzbaşı et al. [36] proposed an improved Legendre method for to obtain the approximate solutions of a class of the integro-differential equations. Wei and Chen [37] presented a numerical method called spectral methods for classes Volterra type integro-differential equations with weakly singular kernel and smooth solutions.
The maximum error for the corrected Jacobi polynomial solution (37) is calculated in a similar way,   and the results are shown in Table 3 for miscellaneous values of N, M. The decrease in maximum error, as M increases, is indisputable. Finally, the third-order FIDDE has also been solved using Legendre, Gegenbauer (also Chebyshev), and Jacobi polynomials, for comparison purposes. The maximum error values are given in Table 4, and it is seen that Jacobi-based solution gives slightly better results.

Example 3
The third example is a second-order FIDE [24][25][26] with variable coefficients under the boundary conditions The exact solution of this problem is  Table 5 and Fig. 2 show a comparison of the absolute error with the corrected absolute errors, for N ¼ 5; 8 and     [25,26] The last example is the second-order Fredholm integro-differential equation The exact solution of problem is Taking ða ¼ 0:4; b ¼ 0:5Þ, the absolute errors of Jacobi polynomial solution for N ¼ 7 and the absolute errors of the improved Jacobi polynomial solution for N ¼ 7; M ¼ 8 are compared with those of the wavelet Galerkin, the wavelet collocation, and the Chebyshev finite difference (ChFD) methods [25,26], in Table 6. Considering the errors of the different methods, it is observed that the smallest errors are obtained using the improved Jacobi polynomial solution. x 3=2 24 À x 2 À Á ; 0 x 1: We assume that the problem has a Jacobi polynomial solution in the form ð Þ ¼ 0:2; À0:3 ð Þ , which are chosen arbitrary. Using the mentioned methods, the Jacobi polynomial solution of the problem is obtained by y 0:2;À0:3 ð Þ 3 x ð Þ ¼ x À x 3 , which is the exact solution of the problem [39]. Furthermore, we can obtain the exact solution of the problem for any value of N ! 3 and corresponding suitable values of a; b ð Þ.

Conclusions
A new matrix method based on Jacobi polynomials and collocation points has been introduced to solve high-order linear FIDDE with variable coefficients. Jacobi polynomials are the common set of orthogonal polynomials, x ð Þ for Example 3 which are the most extensively studied and widely applied systems. The solution of the FIDDE is expressed as a truncated series of orthogonal Jacobi polynomials, which is then transformed from algebraic form into matrix form. The problem and the mixed conditions are also represented in matrix form. Finally, the solution is obtained as a truncated Jacobi series written in matrix form using collocation points. A new error estimation procedure for polynomial solution and a technique to find a high accuracy solution are developed.
Most of the previous studies dealt with solutions using Legendre, Chebyshev, and Gegenbauer polynomials. In this study, however, we have proposed a Jacobi polynomial solution that comprises all of these polynomial solutions.
The new Jacobi matrix method has been applied to four illustrative examples. It is well seen from these examples that the method yields either the exact solution or a high accuracy approximate solution for delay integro-differential equation problems. The accuracy of the approximate solution can be increased using the proposed error analysis technique depending on residual function.
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