Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

. The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.


Introduction
Spectral methods play prominent roles in various fields of applied science such as fluid dynamics.The main idea behind spectral methods is to approximate solutions of differential equations by means of truncated series of orthogonal polynomials, say, ∑     .The three popular techniques employed to determine the expansion coefficients   are the collocation, tau, and Galerkin methods (see, e.g., [1][2][3]).The collocation approach requires the differential equation to be satisfied exactly at the selected collocation points.The tau method is a synonym for expanding the residual function as a series of orthogonal polynomials and then applying the boundary conditions as constraints.The Galerkin approach depends on combining the original basis functions into a new set in which all the functions satisfy the boundary conditions and then enforcing the residual to be orthogonal with the basis functions.The employment of Galerkin techniques is successfully applied on linear problems.For example, in the two papers [4,5], the authors obtained numerical algorithms for solving high even-and high odd-order boundary value problems (BVPs) by applying the Galerkin and Petrov-Galerkin methods.Precisely, they constructed combinations of orthogonal polynomials satisfying the underlying boundary conditions on the given BVP, then applying the Galerkin method on even-order BVPs and a Petrov-Galerkin method on oddorder BVPs for the sake of converting each equation with its boundary conditions to a system of algebraic equations.The suggested algorithms in these articles are suitable for handling linear high-order BVPs.The application of Galerkin and Petrov-Galerkin methods on linear problems has a great advantage that their applications enable one to investigate carefully the resulting systems, especially their complexities and condition numbers.
Many practical problems in various fields of applied science are described by linear or nonlinear boundary value problems.The nonlinear BVPs arise frequently in many areas of science and engineering.Due to the great importance of high-order BVPs, there is an extensive work in literature in the numerical solutions of these problems.In particular, third-order BVPs are important in the area of physics and engineering.For example, some draining or coating fluid flow algorithms for numerically solving the integrated forms of third-and fifth-order differential equations based on employing a dual Petrov-Galerkin method using two new families of general parameters generalized Jacobi polynomials.More recently, Abd-Elhameed et al. in [13] suggested two Legendredual-Petrov-Galerkin algorithms for solving the integrated forms of high odd-order BVPs, while Doha et al. in [14] have developed some algorithms for handling third-and fifthorder linear two point boundary value problems based on nonsymmetric generalized Jacobi Petrov-Galerkin method.We point out here that the algorithms in the two papers [13,14] are capable of handling liner odd-order BVPs with constant coefficients.
It is well-known that the approach of employing operational matrices of differentiation and integration is considered as an important technique for solving many engineering and physical problems.This approach is characterized by its simplicity in application and its capability of handling both linear and nonlinear differential equations.There is a large number of articles in literature in this direction.To the best of our knowledge, all the used operational matrices for handling various types of differential equations are of tau type.For example, the authors in [15] employed the tau operational matrices of derivatives of Chebyshev polynomials of the second kind for handling the singular Lane-Emden type equations.Some other studies in [16][17][18] employ tau operational matrices of derivatives for solving the same type of equations.Other kinds of differential equations were handled by the same technique (see, e.g., [19][20][21][22]).
The main objective of this paper is to introduce a novel Galerkin operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials and then employing it for solving both linear and nonlinear third-order BVPs based on the application of Galerkin and collocation methods.
The contents of the paper are organized as follows.Section 2 is devoted to presenting some preliminaries and relations which will be used throughout the paper.Section 3 is concerned with establishing a Galerkin operational matrix of derivatives of certain generalized Jacobi polynomials.Section 4 is concerned with implementing and presenting two new algorithms for solving linear and nonlinear thirdorder BVPs based on employing the two numerical methods, namely, generalized Jacobi-Galerkin operational matrix method (GJGOMM) for linear problems and the generalized Jacobi collocation operational matrix method (GJCOMM) for nonlinear problems.Also, in this section, the convergence analysis of the used nonsymmetric generalized Jacobi expansion is carefully investigated.Some numerical experiments including some discussions and comparisons are given in Section 5 aiming to illustrate the applicability and efficiency of the suggested algorithms.Finally, some conclusions are reported in Section 6.

Preliminaries
This section is concerned with presenting some definitions, properties, and relations which will be useful throughout this paper.
Theorem 1 (see [13]).If the  times repeated integration of  *  () is denoted by then and  −1 () is a polynomial in  of degree at most ( − 1).
The shifted Jacobi polynomials P(,)  () on [, ] are defined by These polynomials are orthogonal on [, ] with respect to the weight function ( − )  ( − )  , in the sense that Let  , () = ( − )  ( − )  .We denote by  2  , (, ) the weighted  2 space with inner product: and the associated norm ‖‖  , = (, ) 1/2  , .Now, we extend the definition of shifted Jacobi polynomials to include the cases in which  and/or  ≤ −1.Explicitly, if we let , ℓ ∈ Z (the set of all integers), then we define It should be noted here that this definition coincides with the definition introduced by Guo et al. [25] (for the case ).Moreover, an important property of the shifted generalized Jacobi polynomials (SGJPs) is that, for , ℓ ∈ Z + ,

Generalized Jacobi Galerkin Operational Matrix of Derivatives
In this section, a novel Galerkin operational matrix of derivatives will be established.For this purpose, we choose the following set of basis functions: It is easy to see that the set of polynomials {  ():  = 0, 1, 2, . ..} is a linearly independent set.Moreover, these polynomials are orthogonal on [, ] with respect to the weight function () = 1/( − ) 2 ( − ), in the sense that It is not difficult to show that the polynomials   () can be expressed in terms of the shifted Legendre polynomials as Now, we define the space where  2  () is the sobolev space defined in [25].Let   be the space of all polynomials of degree less than or equal to , and set   =  ∩   .

Abstract and Applied Analysis
We observe that Now, if we assume () ∈ , then it has the following expansion in terms of the polynomials   () as where The function () in ( 18) can be approximated by the first ( + 1) terms as where Now, we are going to state and prove the main theorem of this paper, from which a novel Galerkin operational matrix will be introduced.

Solution of Third-Order Two-Point BVPs with Convergence Analysis
In this section, we are interested in developing two new algorithms for solving both linear and nonlinear thirdorder two-point BVPs.The introduced Galerkin operational matrix of derivatives is employed for this purpose.The linear equations are handled by the application of GJGOMM, while the nonlinear equations are handled by the application of GJCOMM.
Remark 4. It should be noted that problem (36), governed by the nonhomogeneous boundary conditions can be easily transformed to a problem similar to (36) and (37) (see, [14]).

Convergence Analysis.
In this section, we investigate the convergence analysis of the suggested expansion.Indeed, in the following, we state and prove a theorem in which the expansion in (18) of a function () = ( − ) 2 ( − )() ∈  (0) (the space defined in ( 16)) converges uniformly to (), under the assumption that the second derivative of the function () is bounded.
()      ⩽ , can be expanded as an infinite sum of the basis given in (18).This series converges uniformly to ().Moreover, the coefficients in (18) satisfy the inequality Proof.If we start with (19), then one can write and in virtue of (15), the coefficients   may be written in the equivalent form If the integrand in the last relation is integrated by parts twice, and making use of Theorem 1, (in case of  = 2), then, for all  ≥ 2, we have where  (2)   () is given by which can be written as and therefore the coefficients   take the form Abstract and Applied Analysis 7 Now, the substitution (2 −  − ) / ( − ) = cos  converts (52) into the form Finally, it is not difficult to show that, for all  ≥ 3, the following inequality holds: and this completes the proof of the theorem.

Numerical Results and Discussions
In this section, the two presented algorithms in Section 4 are applied to solve both linear and nonlinear third-order boundary value problems.As expected, the accuracy increases as the number of terms of the basis expansion increases.
Example 1.Consider the following singulary perturbed linear third-order boundary value problem (see [28]):   subject to the boundary conditions with the analytic solution () = 3 sin(3).
In Table 1, the maximum absolute error  =      −       is listed when GJGOMM is applied for various values of  and .Moreover, Table 2 presents a comparison between the best absolute errors obtained by the application of GJGOMM in case of  = 10, with the best absolute errors obtained by using the method developed in [28].This table shows that our algorithm is more accurate than the method developed in [28].
In Table 3, the maximum absolute errors resulting from the application of GJGOMM for various values of  are displayed, while Table 4 displays a comparison between the best errors obtained by the application of GJGOMM with the best errors resulting from the application of the following methods: (i) sixth-order method (6OM) applied in [11]; (ii) fourth-order methods (4OMs) applied in [11]; (iii) second-order methods (2OMs) applied in [11]; (iv) fourth degree B-spline functions applied in [29]; (v) quintic splines method applied in [30].
In Table 5, we list the maximum absolute errors using GJCOMM for various values of .In this table, we denote  1 ,  2 ,  3 , and  4 by the maximum absolute errors if the selected collocation points are, respectively, the zeros of the shifted Legnedre polynomial  * +1 (), the shifted Chebyshev polynomials of the first and second kinds  * +1 () and  * +1 (), and the shifted nonsymmetirc Jacobi polynomial  (1,2)  +1 ().Moreover, Table 6 displays a comparison between the best errors obtained by the application of GJCOMM with the best errors resulting from the application of the methods mentioned in Example 2. Remark 6.The numerical results displayed in Tables 4 and 6 clarify that the results obtained from the application of our two algorithms are more accurate than those obtained by all the previously mentioned algorithms.

Conclusions
This paper presents a novel operational matrix of derivatives of certain generalized Jacobi polynomials.Two algorithms based on employing the introduced operational matrix together with the application of the two spectral methods, namely, Galerkin and collocation methods, are analyzed for solving linear and nonlinear third-order boundary value problems.To the best of our knowledge, this is the first time a Galerkin operational matrix of derivatives is utilized for handling boundary value problems.The suggested algorithms in this paper are applicable and easy in implementation.Moreover, high accurate solutions are obtained by making use of the two proposed algorithms.The numerical results are convincing and the resulting approximate solutions are very close to the exact ones.

Table 1 :
Maximum absolute errors  for Example 1.

Table 2 :
Comparison between different methods for Example 1.

Table 4 :
Comparison between different methods for Example 2.

Table 6 :
Comparison between different methods for Example 3.