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A new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourthorder boundary value problems
 Waleed M AbdElhameed^{1, 2}Email author,
 Hany M Ahmed^{3, 4} and
 Youssri H Youssri^{2}
https://doi.org/10.1186/s1366201607532
© AbdElhameed et al. 2016
 Received: 22 May 2015
 Accepted: 11 January 2016
 Published: 26 January 2016
Abstract
This paper reports a novel Galerkin operational matrix of derivatives of some generalized Jacobi polynomials. This matrix is utilized for solving fourthorder linear and nonlinear boundary value problems. Two algorithms based on applying Galerkin and collocation spectral methods are developed for obtaining new approximate solutions of linear and nonlinear fourthorder two point boundary value problems. In fact, the key idea for the two proposed algorithms is to convert the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable numerical solvers. The convergence analysis of the suggested generalized Jacobi expansion is carefully discussed. Some illustrative examples are given for the sake of indicating the high accuracy and effectiveness of the two proposed algorithms. The resulting approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by other existing techniques in the literature.
Keywords
 generalized Jacobi polynomials
 Legendre polynomials
 operational matrix
 fourthorder boundary value problems
 Galerkin and collocation methods
MSC
 65M70
 65N35
 35C10
 42C10
1 Introduction
Spectral methods are global methods. The main idea behind spectral methods is to approximate solutions of differential equations by means of truncated series of orthogonal polynomials. The spectral methods play prominent roles in various applications such as fluid dynamics. The three most used versions of spectral methods are: tau, collocation, and Galerkin methods (see for example [1–8]). The choice of the suitable used spectral method suggested for solving the given equation depends certainly on the type of the differential equation and also on the type of the boundary conditions governed by it.
In the collocation approach, the test functions are the Dirac delta functions centered at special collocation points. This approach requires the differential equation to be satisfied exactly at the collocation points. The taumethod is a synonym for expanding the residual function as a series of orthogonal polynomials and then applying the boundary conditions as constraints. The tau approach has an advantage that it can be applied to problems with complicated boundary conditions. In the Galerkin method, the test functions are chosen in a way such that each member of them satisfies the underlying boundary conditions of the given differential equation.
There is extensive work in the literature on the numerical solutions of highorder boundary value problems (BVPs). The great interest in such problems is due to their importance in various fields of applied science. For example, a large number of problems in physics and fluid dynamics are described by problems of this kind. In this respect, there is a huge number of articles handling both high odd and high evenorder BVPs. For example, in the sequence of papers, [5, 9–11], the authors have obtained numerical solutions for evenorder BVPs by applying the Galerkin method. The main idea for obtaining these solutions is to construct suitable basis functions satisfying the underlying boundary conditions on the given differential equation, then applying Galerkin method to convert each equation to a system of algebraic equations. The suggested algorithms in these articles are suitable for handling one and two dimensional linear evenorder BVPs. The Galerkin and PetrovGalerkin methods have the advantage that their applications on linear problems enable one to investigate carefully the resulting systems, especially their complexities and condition numbers.
There are many algorithms in the literature which are applied for handling fourthorder boundary value problems. For example, Bernardi et al. in [12] suggested some spectral approximations for handling two dimensional fourthorder problems. In the two leading articles of Shen [13, 14], the author developed direct solutions of fourthorder two point boundary value problems. The suggested algorithms in these articles are based on constructing compact combinations of Legendre and Chebyshev polynomials together with the application of the Galerkin method. Many other techniques were used for solving fourthorder BVPs, for example, variational iteration method is applied in [15], nonpolynomial sextic spline method in [16], quintic nonpolynomial spline method in [17], and the Galerkin method (see [18, 19]). Theorems which list the conditions for the existence and uniqueness of solution of such problems are thoroughly discussed in the important book of Agarwal [20].
The approach of employing operational matrices of differentiation and integration is considered an important technique for solving various kinds of differential and integral equations. The main advantage of this approach is its simplicity in application and its capability for handling linear differential equations as well as nonlinear differential equations. There are a large number of articles in the literature in this direction. For example, the authors in [6], employed the tau operational matrices of derivatives of Chebyshev polynomials of the second kind for handling the singular LaneEmden type equations. Some other studies in [21, 22] employ tau operational matrices of derivatives for solving the same type of equations. The operational matrices of shifted Chebyshev, shifted Jacobi, and generalized Laguerre polynomials and other kinds of polynomials are employed for solving some fractional problems (see for example, [23–27]). In addition, recently in the two papers of AbdElhameed [28, 29] one introduced and used two Galerkin operational matrices for solving, respectively, the sixthorder two point BVPs and LaneEmden equations.

Establishing a novel Galerkin operational matrix of derivatives of some generalized Jacobi polynomials.

Investigating the convergence analysis of the suggested generalized Jacobi expansion.

Employing the introduced operational matrix of derivatives to numerically solve linear fourthorder BVPs based on the application of Galerkin method.

Employing the introduced operational matrix of derivatives for solving the nonlinear fourthorder BVPs based on the application of collocation method.
The contents of the paper is organized as follows. Section 2 is devoted to presenting an overview on classical Jacobi and generalized Jacobi polynomials. Section 3 is concerned with deriving the Galerkin operational matrix of derivatives of some generalized Jacobi polynomials. In Section 4, we implement and present two numerical algorithms for the sake of handling linear and nonlinear fourthorder BVPs based on the application of generalized JacobiGalerkin operational matrix method (GJGOMM) for linear problems and generalized Jacobi collocation operational matrix method (GJCOMM) for nonlinear problems. Convergence analysis of the generalized Jacobi expansion is discussed in detail in Section 5. Numerical examples including some discussions and comparisons are given in Section 6 for the sake of testing the efficiency, accuracy, and applicability of the suggested algorithms. Finally, conclusions are reported in Section 7.
2 An overview on classical Jacobi and generalized Jacobi polynomials
Lemma 1
Now, Lemma 2 is a direct consequence of Lemma 1.
Lemma 2
The following lemma is also of interest in the sequel.
Lemma 3
3 Generalized Jacobi Galerkin operational matrix of derivatives
Theorem 1
Proof
Corollary 1
4 Two algorithms for fourthorder two point BVPs
In this section, we are interested in developing two numerical algorithms for solving both of the linear and nonlinear fourthorder two point BVPs. The Galerkin operational matrix of derivatives that introduced in Section 3 is employed for this purpose. The linear equations are handled by the application of the Galerkin method, while the nonlinear equations are handled by the application of the typical collocation method.
4.1 Linear fourthorder BVPs
4.2 Solution of nonlinear fourthorder two point BVPs
5 Convergence analysis of the approximate expansion
In this section, the convergence analysis of the suggested generalized Jacobi approximate solution will be investigated. We will state and prove a theorem in which the expansion in (12) of a function \(f(x)=(xa)^{2} (bx)^{2} G(x)\in H_{0,w}^{2}(I)\), where \(G(x)\) is of bounded fourth derivative, converges uniformly to \(f(x)\).
Theorem 2
Proof
6 Numerical results and discussions
In this section, the two proposed algorithms in Section 4 are applied to solve linear and nonlinear fourthorder two point boundary value problems. The numerical results ensure that the two algorithms are very efficient and accurate.
Example 1
Maximum absolute error of \(\pmb{yy_{N}}\) for Example 1
N  E 

4  1.283929 × 10^{−5} 
6  2.28788 × 10^{−8} 
8  3.45852 × 10^{−11} 
10  4.17094 × 10^{−14} 
12  2.03407 × 10^{−15} 
14  1.96477 × 10^{−15} 
Comparison between the relative errors for Example 1
x  1OM [35]  2OMs [35]  GJGOMM ( N = 8)  GJGOMM ( N = 12) 

0.25  5.39791 × 10^{−3}  8.11851 × 10^{−3}  1.09077 × 10^{−10}  2.75173 × 10^{−15} 
0.50  1.57835 × 10^{−2}  2.09205 × 10^{−2}  1.14685 × 10^{−10}  1.33747 × 10^{−15} 
0.75  2.54797 × 10^{−2}  2.93281 × 10^{−2}  8.24739 × 10^{−11}  4.50799 × 10^{−16} 
1.00  3.14713 × 10^{−2}  3.09264 × 10^{−2}  3.3689 × 10^{−12}  5.37406 × 10^{−16} 
1.25  3.22814 × 10^{−2}  2.65317 × 10^{−2}  7.75226 × 10^{−11}  2.27451 × 10^{−16} 
1.50  2.73173 × 10^{−2}  1.82938 × 10^{−2}  9.08649 × 10^{−11}  2.54606 × 10^{−15} 
1.75  1.64910 × 10^{−2}  8.68533 × 10^{−3}  7.53861 × 10^{−11}  9.75198 × 10^{−16} 
Example 2
Maximum absolute error of \(\pmb{yy_{N}}\) for Example 2 for \(\pmb{N=2,4,6,8,10,12}\)
N  \(\boldsymbol {E_{1}}\)  \(\boldsymbol {E_{2}}\)  \(\boldsymbol {E_{3}}\)  \(\boldsymbol {E_{4}}\) 

2  1.36037 × 10^{−6}  2.11456 × 10^{−6}  8.70645 × 10^{−7}  8.46647 × 10^{−8} 
4  1.23719 × 10^{−8}  2.69013 × 10^{−8}  5.15533 × 10^{−9}  5.83994 × 10^{−10} 
6  4.34874 × 10^{−11}  1.19637 × 10^{−10}  1.93656 × 10^{−11}  4.17821 × 10^{−12} 
8  2.65565 × 10^{−13}  4.8217 × 10^{−13}  1.57874 × 10^{−13}  3.44169 × 10^{−14} 
10  9.10383 × 10^{−15}  9.76996 × 10^{−15}  1.19904 × 10^{−14}  8.88178 × 10^{−15} 
12  8.65974 × 10^{−15}  8.65974 × 10^{−15}  1.06581 × 10^{−14}  8.43769 × 10^{−15} 
Comparison between the absolute errors for Example 2
x  Method in [36]  Method in [37] ( \(\boldsymbol {\phi_{2}y}\) )  Method in [37] ( \(\boldsymbol {\psi_{2}y}\) )  GJCOMM ( N = 4) 

0.0  0.0  0.0  0.0  2.55351 × 10^{−15} 
0.2  1.79 × 10^{−4}  4.1883 × 10^{−4}  4.1012 × 10^{−6}  6.94449 × 10^{−8} 
0.4  3.25 × 10^{−4}  1.2786 × 10^{−3}  1.2574 × 10^{−5}  6.30457 × 10^{−8} 
0.6  4.08 × 10^{−4}  1.9971 × 10^{−3}  1.9762 × 10^{−5}  4.48952 × 10^{−8} 
0.8  4.02 × 10^{−4}  2.0753 × 10^{−3}  2.0714 × 10^{−5}  6.49632 × 10^{−8} 
1.0  2.94 × 10^{−4}  1.3038 × 10^{−3}  1.3163 × 10^{−5}  1.47025 × 10^{−8} 
1.2    1.8581 × 10^{−4}  1.9026 × 10^{−6}  2.10697 × 10^{−8} 
4e  2 × 10^{−9}  2.4400 × 10^{−15}  0  7.77156 × 10^{−16} 
Example 3
Maximum absolute error of \(\pmb{yy_{N}}\) for Example 3 for \(\pmb{N=2,4,6}\)
N  E 

2  5.489 × 10^{−17} 
4  2.265 × 10^{−17} 
6  2.10662 × 10^{−17} 
Comparison between the absolute errors for Example 3
x  Method in [37] ( \(\boldsymbol {\psi_{2}y}\) )  Method in [37] ( \(\boldsymbol {\phi_{2}y}\) )  GJCOMM ( N = 2) 

0.0  0.0  0.0  0.0 
0.2  8.1093 × 10^{−10}  3.5906 × 10^{−5}  2.03045 × 10^{−18} 
0.4  2.0542 × 10^{−9}  1.0188 × 10^{−4}  9.04773 × 10^{−18} 
0.6  2.2272 × 10^{−9}  1.3579 × 10^{−4}  2.10718 × 10^{−17} 
0.8  1.0115 × 10^{−9}  8.5908 × 10^{−5}  3.74696 × 10^{−17} 
1.0  0.0  5.5799 × 10^{−13}  5.489 × 10^{−17} 
Example 4
Comparison between the absolute errors for Example 4
x  GJCOMM ( N = 4)  GJCOMM ( N = 6)  GJCOMM ( N = 8)  RKHSM ( \(\boldsymbol {u^{101}_{1}}\) ) [38] 

0.0  0.0  0.0  0.0  0.0 
0.1  1.74774 × 10^{−11}  2.42005 × 10^{−14}  9.34812 × 10^{−17}  2.78 × 10^{−8} 
0.2  8.7269 × 10^{−11}  1.5889 × 10^{−14}  8.29415 × 10^{−18}  8.09 × 10^{−8} 
0.3  2.27345 × 10^{−11}  3.59562 × 10^{−14}  2.17857 × 10^{−18}  1.20 × 10^{−7} 
0.4  1.213 × 10^{−10}  6.36858 × 10^{−14}  6.03901 × 10^{−17}  1.25 × 10^{−7} 
0.5  6.32272 × 10^{−12}  1.87133 × 10^{−15}  2.06595 × 10^{−16}  9.56 × 10^{−8} 
0.6  1.27768 × 10^{−10}  6.44606 × 10^{−14}  1.21431 × 10^{−17}  4.82 × 10^{−8} 
0.7  1.44017 × 10^{−11}  4.09586 × 10^{−14}  1.10589 × 10^{−16}  7.38 × 10^{−9} 
0.8  9.92086 × 10^{−11}  1.56333 × 10^{−14}  8.74301 × 10^{−16}  1.07 × 10^{−8} 
0.9  2.37061 × 10^{−11}  2.57225 × 10^{−14}  3.38271 × 10^{−17}  7.08 × 10^{−9} 
1.0  4.44089 × 10^{−16}  2.70617 × 10^{−16}  2.27249 × 10^{−16}  0.0 
7 Concluding remarks
In this article, a novel operational matrix of derivatives of certain generalized Jacobi polynomials is derived and used for introducing spectral solutions of linear and nonlinear fourthorder two point boundary value problems. The two spectral methods, namely the Galerkin and collocation methods are employed for this purpose. The main advantages of the introduced algorithms are their simplicity in application, and also their high accuracy, since highly accurate approximate solutions can be achieved by using a small number of terms of the suggested expansion. The numerical results are convincing and the resulting approximate solutions are very close to the exact ones.
Declarations
Acknowledgements
The authors are grateful to the referees for their valuable comments and suggestions which have improved the manuscript in its present form.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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