Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

Eid H. Doha, Mohamed A. Abdelkawy, Ahmed Z.M. Amin, Dumitru Baleanu Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt eiddoha@sci.cu.edu.eg Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia melkawy@yahoo.com Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Department of Basic Science, Institute of Engineering, Canadian International College (CIC), Giza, Egypt azm.amin@yahoo.com Department of Mathematics, Cankaya University, Ankara, Turkey dumitru@cankaya.edu.tr f Institute of Space Sciences, Magurele-Bucharest, Romania


Introduction
Recently, the studies of IDEs were developed very intensively and speedily.Numerical solutions of the IDEs have received considerable attention not only in mathematics, but also in computational physics.These equations are combinations of the unknown functions that appear under the sign of integration and derivatives.In addition, IDEs are used c Vilnius University, 2019 in many problems of mechanics, engineering, chemistry, physics, biology, astronomy, potential theory, electrostatics, etc. [3, 12, 22, 25, 31-33, 37-39, 45].
In few years, many published papers (e.g., [13,16,31,35]) were devoted for solving IDEs.Maleknejad et al. [30] implemented the Bernstein operational matrix to study a system of linear Volterra-Fredholm IDEs.The nonlinear IDEs were studied by means of the meshless method in [17].Zarebnia [47] developed and proposed an efficacious numerical solution for the Volterra IDEs by using sinc method.Recently, variational iteration method were applied by Nadjafi et al. [36] to numerically solve the system of IDEs.More recently, Yüzbaşı [46] proposed the collocation approach for solving Fredholm-Volterra IDEs.
In last years, there are high level of interest of the spectral methods for solving many kinds of differential and IDEs due to their ease of application for finite and infinite domains [1,2,28,34,40,44].The speed of convergence is one of the great feature of the spectral method.Besides, the spectral methods have enormous rates of convergence, they also have high level of reliability.The spectral method were divided into four classifications: collocation [5,6,26,27], tau [8,20,24,41], Galerkin [14,15,23] and Petrov-Galerkin [4,29] method.The main idea of the spectral methods is to express the solution of the problem as a finite sum of given basis of functions (orthogonal polynomials or combination of orthogonal polynomials) and then to choose the coefficients in order to minimize the difference between the exact and the numerical solutions.
Our motivation in this paper is to develop spectral approximation of IDEs.We propose the shifted Jacobi-Gauss collocation (SJGC) method to find the solution u N (x) by means of the shifted Jacobi polynomial.The IDEs is collocated at the selected points.For convenient, we use the nodes of the shifted Jacobi-Gauss (SJG) interpolation as collocation points.These equations together with the initial conditions produces a system of linear algebraic equations, which can be easily solved.This scheme is one of the most suitable methods for solving system of algebraic equations.
The outlines of the present paper are arranged as follows.We present few revelent properties of shifted Jacobi polynomials in the following section.In Section 3, we propose the SJGC scheme to solve one-dimensional IDEs.In Section 4, we solve linear twodimensional Volterra IDEs with the initial conditions.While in Section 5, we present some useful lemmas and error analysis of the IDEs.In Section 6, several numerical examples and comparisons between our numerical results and those of other methods are discussed.Finally, Section 7 outlines the conclusions.

Properties of shifted Jacobi polynomials
By means of the main properties of Jacobi polynomials, we conclude the following: where Nonlinear Anal.Model.Control, 24(3):332-352 Furthermore, the rth derivative of P (α,β) j (x) is computed as where r is an integer.Let the shifted Jacobi polynomial Taking w x β , we list the following inner product and norm related to the weighted space L 2 [0, L]-orthogonal system is consisted of a set of shifted Jacobi polynomials, where where [7,9,18] h . https://www.mii.vu.lt/NA We used x (α,β) N,j and (α,β) N,j , 0 j N , as the nodes and Christoffel numbers of the standard Jacobi-Gauss interpolation in the interval [−1, 1].For shifted Jacobi-Gauss interpolation on [0, L], we find For any positive integer N , φ ∈ S 2N +1 [0, L] and by means of Jacobi-Gauss quadrature property, L,N,j .

Volterra IDEs with the initial condition
In this subsection, we use the spectral collocation method to solve the following Volterra IDEs: subject to where k(x, s) and f (x) are given functions and γ i (i = 0, . . ., m) are constants, while u(x) is unknown function.We using the SJGC algorithm to transform the previous IDEs into system of algebraic equations.Thus, we approximate the independent variable using the SJGC algorithm at x (θ,ϑ) L,N,j nodes.The nodes are the set of points in a specified domain, where the dependent variable values are approximated.In general, the choice of the location of the nodes are optional, but taking the roots of the shifted Jacobi polynomials referred to as SJG points, gives particularly accurate solutions for the spectral methods.Now, we outline the main step of the SJGC method for solving one-dimensional IDEs.We choose the approximate solution to be of the form Then where Equation ( 3) can be written as In the proposed SJGC method, the residual of ( 6) is set to zero at (N − m + 1) of SJG points.Then, adopting ( 5)-( 6), we can be write (3) in form: L,j (s) ds , n = m, . . ., N.

System of Volterra IDEs with the initial conditions
In the current subsection, we apply the technique discussed in Section 3.1 to solve system of Volterra IDEs in the form subject to where f 1 (x), f 2 (x), k 1 (x, t) and k 2 (x, t) are given function, and γ i , ζ i (i = 0, . . ., m) are constants, while u(x), v(x) are unknown functions.
In the SJGC method, the approximate solution can be introduced as a truncated shifted Jacobi series: and, in virtue of (11), we deduce that In the proposed SJGC method, the residual of ( 9) is set to be zero at 2(N − m + 1) of SJG points, thus we find Nonlinear Anal.Model.Control, 24(3):332-352 where n = m, . . ., N .Using (2), Eq. ( 10) can be reformulated as Equations ( 12) and (13) give a system of algebraic equations, which can be solved for the unknown coefficients a j and b j .So, u N (x) and v N (x) given in Eq. ( 11) can be estimated.

Mixed of Volterra-Fredholm IDEs with the initial conditions
This section present an efficient spectral algorithm by means of the SJGC method to numerically solve linear mixed Volterra-Fredholm IDEs in the form subject to Similar steps to that given in the previous subsections, enable one to write Eq. (3.3) in the form L,j (s) ds Based on the information in the previous section, we get the following system of algebraic equations: https://www.mii.vu.lt/NAThe previous linear system of algebraic equations can be easily solved.After determining the coefficients a j , it is straightforward to compute the approximate solution u N (x) at any value of x ∈ [0, L] in the given domain from the following equation 4 Two-dimensional IDEs In this section, we extend the above analysis to solve the following two-dimensional linear Volterra IDEs: subject to where k(x, t, y, z) and f (x, t) are given functions, while u(x, t) is unknown function.Therefore, the SJGC method will be applied to transform the previous two-dimensional Volterra IDEs into system of algebraic equations.
Let us expand the dependent variable in the form The partial derivatives of the approximate solution u N,M (x, t) is then estimated as where , .

Lemmas and error analysis
Some useful lemmas and a discussion about the error analysis of the algorithm presented in Section 3.1.

Lemmas
Definition 1.Let P N : L 2 (I) → X N be the L 2 orthogonal projection, defined by Definition 2. Some weighted Hilbert spaces will be presented here.For a nonnegative integer m, define [10,11,42] ), 0 i m , where ∂ i x υ(x) = ∂ i υ(x)/∂x i related to the following seminorm and the norm: ).The interpolation of u (I α,β N u) computed at any points of Jacobi-Gauss points (Gauss or Gauss-Radau or Gauss-Lobatto points) stasifies the following estimates [11]:

Error analysis
The main goal is to estimate the accuracy of the solutions we obtained.In this section, error analysis for the introduced technique (3) will be discussed.We provided error analysis for the proposed method to indicate its exponential rate of convergence, provided that the source and kernel functions are sufficiently smooth.In order to do that, some properties of Banach algebras and Sobolev inequality are taken into account.

Numerical results
We listed several examples to illustrate the powerful and effectiveness of the proposed method.The mentioned comparisons of the numerical results detect that the previous algorithms are very appropriate and effective.
The difference between the exact solution and the value of the approximate solution is define as the absolute error (AE) given by where u(x) and u N (x) are the exact solution and the approximate solutions at the point x, respectively.Moreover, the maximum absolute errors (MAE) is given by Example 1. Firstly, we introduce the linear Volterra IDEs in the form [47] with the initial condition u(0) = 0, knowing that the exact solution given by u(x) = ln(x + 1).
A comparison between the MAE using the proposed method and the sinc method [47] is summarized in Table 1 with several choices of θ and ϑ.The numerical results presented in the Table 1 show that results are vary accurate for small value of N .
Figure 1 compares graphically the curves of numerical and exact solutions of problem (1).Moreover, we represent the logarithmic graphs of M E (i.e., log 10 M E ) obtained by the novel algorithm with different values of N in Fig. 2.   Example 2. Let us, consider the system Volterra IDEs in the form [30] with the condition knowing that the exact solution given by u 1 (x) = e x and u 2 (x) = 1 + sin(x).
https://www.mii.vu.lt/NAApplying the technique described in Section 3.2 with different choice of N , the present method is more accurate than Bernstein operational matrix [30], see Table 2.The curves of the AE E 1 (AE of u 1 ) and E 2 (x) (AE of u 2 ) of Example 2 for N = 12, are displayed in Figs. 3 and 4, respectively.In Fig. 5, we depict the logarithmic graphs of the MAE (i.e., log 10 M 1 , log 10 M 2 ) for various values of N .This demonstrates that the new algorithm provides accuracy approximation and product exponential convergence rates.
knowing that the exact solution is given by u(x) = e x .In order to confirm high accuracy of the novel algorithm for mixed Volterra-Fredholm IDEs problem, Table 3 introduce a comparison between the maximum absolute errors obtained in [46] and the results obtained in this paper with various choices of N .We observed that a good approximation of the mixed Volterra-Fredholm IDEs is achieved for small of N .
In Fig. 6, we see the matching of the value of the AE in these figures and find in Table 3.Moreover, we represent the logarithmic graphs of M E (i.e., log 10 M E ) obtained by the proposed method with several values of N in Fig. 7.   where (x, t) ∈ [0, 1] × [0, 1], subject to u(x, 0) = 0, ∂u ∂t (x, 0) = x, the f (x, t) is given such that the exact solution is u(x) = x sin(t).
Table 4 lists the AE for several choices of N and M of Example 4. We observed a good approximation of linear two-dimensional space Volterra IDEs.The AE for Example 4 was displayed in Fig. 8 for N = M = 8.

Table 1 .
The MAE for Example 1.

Table 2 .
The AE for Example 2.