A new exponential Jacobi pseudospectral method for solving high-order ordinary differential equations
- Ali H Bhrawy^{1, 2}Email author,
- Ramy M Hafez^{3} and
- Jameel F Alzaidy^{2}
https://doi.org/10.1186/s13662-015-0491-x
© Bhrawy et al.; licensee Springer. 2015
Received: 27 March 2015
Accepted: 29 April 2015
Published: 14 May 2015
Abstract
This paper reports new orthogonal functions on the half line based on the definition of the classical Jacobi polynomials. We derive an operational matrix representation for the differentiation of exponential Jacobi functions which is used to create a new exponential Jacobi pseudospectral method based on the operational matrix of exponential Jacobi functions. This exponential Jacobi pseudospectral method is implemented to approximate solutions to high-order ordinary differential equations (ODEs) on semi-infinite intervals. The advantages of using the exponential Jacobi pseudospectral method over other techniques are discussed. Several numerical examples are presented to confirm the validity and applicability of the proposed method. Moreover, the obtained results are compared with those obtained using other techniques.
Keywords
high-order ODEs exponential Jacobi functions operational matrix of differentiation pseudospectral method1 Introduction
In this article, we derive the operational matrix of differentiation of exponential Jacobi functions, and then we implement a new exponential Jacobi pseudospectral method in conjunction with the operational matrix of differentiation of exponential Jacobi functions to obtain numerical solutions of high-order ordinary differential equations on a semi-infinite interval. Spectral methods (see, for instance, [25–29]), based on using operational matrices, have been implemented in various problems such as fractional differential equations [30, 31], fractional optimal control problems [32], Lane-Emden equation [33, 34], and various integral equations [35]. This equation is collocated at the exponential Jacobi-Gauss quadrature nodes. Doing so, we find that we can obtain very accurate results with minimal computation. Hence, the method is rather computationally efficient compared with other numerical or analytical approaches. Finally, numerical experiments of high-order ODEs are implemented to demonstrate the validity and efficiency of the proposed algorithm. In particular, in Section 2 we design the exponential Jacobi pseudospectral method technique for solving high-order ODEs. In Section 3, several numerical examples are presented to demonstrate the efficiency of present numerical algorithm. Finally, in Section 4, a few concluding remarks and future work are included.
2 Exponential Jacobi pseudospectral method
This section presents technical details of the new exponential Jacobi functions and the exponential Jacobi pseudospectral method (EJPM). First, we outline some useful mathematical properties that we shall make use of. Then, the derivative operational matrix of exponential Jacobi functions is derived and proved. Finally, we derive the EJPM.
2.1 Mathematical preliminaries
Here, we outline the exponential Jacobi-Gauss quadrature. Assume that \(x^{(\theta,\vartheta)}_{N,j}\), \(0\leqslant j\leqslant N\), are the zeros of the Jacobi-Gauss interpolation on the interval \((-1, 1)\) and \(\varpi^{(\theta,\vartheta)}_{N,j}\), \(0\leqslant j\leqslant N\), are the corresponding weights of this interpolation. The nodes of the exponential Jacobi-Gauss interpolation on the interval \((0,\infty)\) are the zeros of \(EJ_{N+1}^{(\theta, \vartheta)}(x)\), which are denoted by \(x^{(\theta,\vartheta)}_{R,N,j}\), \(0\leqslant j\leqslant N\). Clearly \(x^{(\theta,\vartheta)}_{R,N,j} =-L \ln(\frac{1-x^{(\theta,\vartheta)}_{N,j}}{2})\), and weights are \(\varpi^{(\theta,\vartheta)}_{R,N,j} =\frac{1}{2^{\theta+\vartheta+1}} \varpi^{(\theta,\vartheta)}_{N,j}\), \(0\leqslant j\leqslant N\). Let \(S_{N}(0,\infty)\) be the set of all polynomials of degree at most N.
2.2 The derivative operational matrix of exponential Jacobi function
Here we shall give the derivation of a new operational matrix of derivative of the exponential Jacobi functions, which is essential to our numerical method.
Theorem 2.1
Proof
The main advantages of studying the general class of exponential Jacobi functions is that the exponential Legendre functions, exponential Chebyshev functions of all kinds, and the exponential Gegenbauer functions can be obtained as immediately special cases of the exponential Jacobi functions. Accordingly, in this article we cover all the previous mentioned functions. More specifically, exponential Legendre, exponential Chebyshev and exponential Gegenbauer operational matrices can be obtained as special cases from the derived exponential Jacobi functions. These cases are summarized in the following corollaries.
Corollary 2.2
Corollary 2.3
Corollary 2.4
Corollary 2.5
Corollary 2.6
Remark 2.7
2.3 Derivation of the pseudospectral method
We will obtain a system of \(N+1\) algebraic equations from: (i) applying the operational matrix of an exponential Jacobi function; (ii) collocation of the high-order ordinary differential equations at \((N -m- 1)\) exponential Jacobi-Gauss points; (iii) imposition of m initial conditions.
3 Numerical results
This section presents several numerical examples to demonstrate the high accuracy and applicability of the present method, and all of them were performed on the computer using a program written in Mathematica 8.0. The absolute errors in the given tables are the values of \(|u(x)-u_{N}(x)|\) at selected points. Moreover, the obtained results are compared with those obtained using other techniques. We consider the following examples.
Example 1
Comparison of the absolute errors for Example 1
x | Analytical | ChNN [ 38 ] | EJOM ( N = 20, L = 1) | ||
---|---|---|---|---|---|
\(\boldsymbol {\theta=\vartheta=-\frac{1}{2}}\) | θ = ϑ = 0 | \(\boldsymbol {\theta=\vartheta=\frac{1}{2}}\) | |||
0.0 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 |
0.1 | 0.99004983 | 0.99004883 | 0.99004988 | 0.99004994 | 0.99004993 |
0.2 | 0.96078943 | 0.96077941 | 0.96078946 | 0.96078942 | 0.96078934 |
0.3 | 0.91393118 | 0.9139317 | 0.91393112 | 0.91393112 | 0.91393120 |
0.4 | 0.85214378 | 0.85224279 | 0.85214382 | 0.85214381 | 0.85214376 |
0.5 | 0.77880078 | 0.77870077 | 0.77880085 | 0.77880096 | 0.77880099 |
0.6 | 0.69767632 | 0.69767719 | 0.69767618 | 0.69767594 | 0.69767596 |
0.7 | 0.61262639 | 0.61272838 | 0.61262644 | 0.61262662 | 0.61262661 |
0.8 | 0.527292424 | 0.52729340 | 0.52729252 | 0.52729272 | 0.52729269 |
0.9 | 0.44485806 | 0.44490806 | 0.44485807 | 0.44485772 | 0.44485769 |
1.0 | 0.36787944 | 0.36782729 | 0.36787939 | 0.36787915 | 0.36787921 |
Example 2
Comparison of the absolute errors for Example 2
x | \(\boldsymbol {\theta = \vartheta=\frac{-1}{2}}\) | θ = ϑ = 0 | \(\boldsymbol {\theta = \vartheta=\frac{1}{2}}\) |
---|---|---|---|
0.0 | 0 | 0 | 0 |
1.0 | 3.007.10^{−8} | 2.107.10^{−8} | 2.707.10^{−8} |
2.0 | 2.353.10^{−8} | 4.976.10^{−8} | 5.067.10^{−8} |
3.0 | 3.180.10^{−8} | 5.733.10^{−8} | 5.028.10^{−8} |
4.0 | 9.386.10^{−8} | 2.368.10^{−8} | 6.252.10^{−9} |
5.0 | 9.514.10^{−9} | 6.890.10^{−9} | 1.667.10^{−8} |
6.0 | 1.007.10^{−9} | 4.581.10^{−9} | 3.294.10^{−9} |
7.0 | 6.591.10^{−10} | 1.593.10^{−10} | 9.366.10^{−9} |
8.0 | 1.070.10^{−9} | 3.543.10^{−9} | 1.079.10^{−8} |
9.0 | 1.375.10^{−9} | 5.259.10^{−9} | 1.136.10^{−8} |
10.0 | 6.492.10^{−10} | 5.976.10^{−9} | 1.153.10^{−8} |
Example 3
Comparison of the absolute errors for Example 3
x | RCC [ 36 ] method | EJOM ( N = 7, L = 10) | ||
---|---|---|---|---|
\(\boldsymbol {\theta=\vartheta=-\frac{1}{2}}\) | θ = ϑ = 0 | \(\boldsymbol {\theta=\vartheta=\frac{1}{2}}\) | ||
0.0 | 0 | 0 | 0 | 0 |
0.1 | 1.253.10^{−4} | 2.915.10^{−5} | 3.871.10^{−5} | 3.985.10^{−5} |
0.2 | 5.855.10^{−4} | 1.836.10^{−4} | 2.508.10^{−4} | 2.552.10^{−4} |
0.3 | 1.131.10^{−3} | 4.957.10^{−4} | 6.943.10^{−4} | 6.952.10^{−4} |
0.4 | 2.276.10^{−3} | 9.589.10^{−4} | 1.368.10^{−3} | 1.341.10^{−3} |
0.5 | 3.488.10^{−3} | 1.564.10^{−3} | 2.255.10^{−3} | 2.152.10^{−3} |
0.6 | 4.950.10^{−3} | 2.320.10^{−3} | 3.342.10^{−3} | 3.088.10^{−3} |
0.7 | 6.665.10^{−3} | 3.255.10^{−3} | 4.631.10^{−3} | 4.123.10^{−3} |
0.8 | 8.596.10^{−3} | 4.416.10^{−3} | 6.139.10^{−3} | 5.246.10^{−3} |
0.9 | 1.074.10^{−2} | 5.872.10^{−3} | 7.897.10^{−3} | 6.463.10^{−3} |
1.0 | 1.309.10^{−2} | 7.701.10^{−3} | 9.949.10^{−3} | 7.795.10^{−3} |
Example 4
Comparison of the absolute errors for Example 4
x | θ = ϑ = 0 | \(\boldsymbol {\theta = \vartheta=\frac{1}{2}}\) | θ = ϑ = 1 |
---|---|---|---|
0.0 | 0 | 0 | 0 |
10.0 | 5.810.10^{−8} | 6.143.10^{−8} | 6.856.10^{−8} |
20.0 | 5.880.10^{−8} | 1.450.10^{−6} | 1.264.10^{−6} |
30.0 | 2.953.10^{−7} | 1.094.10^{−6} | 5.662.10^{−8} |
40.0 | 1.933.10^{−7} | 8.744.10^{−7} | 2.319.10^{−6} |
50.0 | 1.047.10^{−7} | 1.617.10^{−6} | 3.126.10^{−6} |
60.0 | 8.421.10^{−7} | 1.775.10^{−6} | 3.296.10^{−6} |
70.0 | 8.012.10^{−8} | 1.805.10^{−6} | 3.329.10^{−6} |
80.0 | 7.940.10^{−8} | 1.811.10^{−6} | 3.335.10^{−6} |
90.0 | 7.924.10^{−8} | 1.812.10^{−6} | 3.336.10^{−6} |
100.0 | 7.924.10^{−8} | 1.812.10^{−6} | 3.336.10^{−6} |
4 Conclusions
In this paper, we derived the operational matrix of derivative of exponential Jacobi functions. This operational matrix in conjunction with the exponential Jacobi spectral collocation method is utilized for reducing the solution of high-order ordinary differential equations on the semi-infinite interval to that of a system of algebraic equations, which may then be solved much more easily. The operational matrices of derivatives of exponential Legendre and exponential Chebyshev functions of the first and second kinds, which often appear in conjunction with such spectral methods in the literature, may be obtained as special cases of the operational matrix of exponential Jacobi functions by taking the corresponding spacial cases of the exponential Jacobi functions parameters θ and ϑ.
Illustrative numerical examples with the satisfactory approximate solutions are achieved to demonstrate the applicability and high accuracy of the present technique. The obtained approximations of the exact solutions for the test problems make this technique very attractive and contributed to the good agreement between approximate and exact values in the numerical example. In addition, the present method could prove fruitful for those investigating not only high-order ordinary differential equations, but more broadly equations with (i) strong nonlinearity and (ii) singularities.
It can be expected that the new exponential Jacobi pseudospectral scheme coupled with a spectral element method will be an effective tool for the numerical solution of time-dependent differential equations [39]. It also may be extended to solve nonlocal boundary value problems with more complicated conditions, meanwhile its extension to the two-dimensional problems is straightforward. We assert that the proposed technique can be applied to a much larger class of fixed-order and variable-order fractional differential equations (see, for instance, [40, 41]).
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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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