Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems
Abstract
:1. Introduction
2. Sinc Function Approximation
2.1. Sinc Function Properties
2.2. The General Sinc–Galerkin Method
2.3. The Sinc Methodology
2.4. Newton’s Method
3. The Variational Iteration Method
3.1. Analysis of the VIM
- convert the truncated series obtained by VIM by using Laplace transform.
- approximate the result that we get from previous step using the Padé approximant.
- convert the output function that we get from the previous step using inverse Laplace transform.
3.2. Adomian—Variational Iteration Method(AVIM)
3.3. Lagrange Multiplier for Special Kind of Equations
4. Numerical Results
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Chanane, B. Computing the eigenvalues of singular Sturm-Liouville problems using the regularized sampling method. Appl. Math. Comput. 2007, 184, 972–978. [Google Scholar] [CrossRef]
- Bender, C.M.; Orszag, S.A. Advanced Mathematical Methods for Scientific and Engineers; McGraw-Hill International Editions; McGraw-Hill: New York, NY, USA, 1987. [Google Scholar]
- Celik, I. Approximate calculation of eigenvalues with the method of weighted residuals-collocation method. Appl. Math. Comput. 2005, 160, 401–410. [Google Scholar]
- Guseinov, G.S.; Karaca, I.Y. Instability intervals of a Hill’s equation with piecewise constant and alternating coefficient. Comput. Math. Appl. 2004, 47, 319–326. [Google Scholar] [CrossRef] [Green Version]
- El-Gamel, M.; Zayed, A.I. Sinc-Galerkin method for solving nonlinear boundary-value problems. Comput. Math. Appl. 2004, 48, 1285–1298. [Google Scholar] [CrossRef]
- Saadatmandi, A.; Razzaghi, M.; Dehghan, M. Sinc-Galerkin solution for nonlinear two-point boundary value problems with applications to chemical reactor theory. Math. Comput. Model. 2005, 42, 1237–1244. [Google Scholar] [CrossRef]
- Stenger, F. A Sinc-Galerkin method of solution of boundary value problems. Math. Comput. 1979, 33, 85–109. [Google Scholar]
- Mohsen, A.; El-Gamel, M. On the Galerkin and collocation methods for two-point boundary value problems using sinc bases. Comput. Math. Appl. 2008, in press. [Google Scholar] [CrossRef]
- Alquran, M.; Al-Khaled, K. Approximations of Sturm-Liouville eigenvalues using sinc-Galerkin and differential transform methods. Appl. Appl. Math. 2010, 5, 128–147. [Google Scholar]
- He, J.H. Variational iteration method—A kind of non-linear analytical technique: Some examples. Int. J. Non-Linear Mech. 1999, 34, 699–708. [Google Scholar] [CrossRef]
- He, J.H. Variational iteration method?some recent results and new interpretations. J. Comput. Appl. Math. 2007, 207, 3–17. [Google Scholar] [CrossRef] [Green Version]
- He, J.H.; Wu, X.H. Variational iteration method: New development and applications. Comput. Math. Appl. 2007, 54, 881–894. [Google Scholar] [CrossRef]
- Abassy, T.A.; El-Tawil, M.A.; El Zoheiry, H. Toward a modified variational iteration method. J. Comput. Appl. Math. 2007, 207, 137–147. [Google Scholar] [CrossRef]
- Jin, L. Application of modified variational iteration method to the Bratu-type problems. Int. J. Contemp Math. Sci. 2010, 5, 153–158. [Google Scholar]
- Islam, S.U.; Haq, S.; Ali, J. Numerical solution of special 12th-order boundary value problems using differential transform method. Commun. Nonlinear Sci. Numer. Simul. 2009, in press. [Google Scholar] [CrossRef]
- Abbasbandy, S. A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. J. Comput. Appl. Math. 2007, 207, 59–63. [Google Scholar] [CrossRef] [Green Version]
- Wazwaz, A.M. The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Appl. Math. Comput. 2007, 188, 485–491. [Google Scholar] [CrossRef]
- Mukhtarov, O.S.; Yücel, M. A Study of the Eigenfunctions of the Singular Sturm-Liouville Problem Using the Analytical Method and the Decomposition Technique. Mathematics 2020, 8, 415. [Google Scholar] [CrossRef] [Green Version]
- Qadir, R.R.; Jwamer, K.H.F. Refinement Asymptotic Formulas of Eigenvalues and Eigenfunctions of a Fourth Order Linear Differential Operator with Transmission Condition and Discontinuous Weight Function. Symmetry 2019, 11, 1060. [Google Scholar] [CrossRef] [Green Version]
- Khashshan, M.M.; Syam, M.I.; Al Mokhmari, A. A Reliable Method for Solving Fractional Sturm-Liouville Problems. Mathematics 2018, 6, 176. [Google Scholar] [CrossRef]
- Alsaedi, A.; Alsulami, M.; Srivastava, H.M.; Ahmad, B.; Ntouyas, S.K. Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions. Symmetry 2019, 11, 281. [Google Scholar] [CrossRef] [Green Version]
- Stenger, F. Numerical Methods Based on Sinc and Analytic Functions; Springer: New York, NY, USA, 1993. [Google Scholar]
- Eggert, N.; Jarratt, M.; Lund, J. Sinc function computation of the eigenvalues of Sturm-Liouville problems. J. Comput. Phys. 1987, 69, 209–229. [Google Scholar] [CrossRef]
- McArthur, K.M.; Arthur, K.M. A Collocative Variation of the Sinc-Galerkin Method for Second Order Boundary Value Problems. In Computation and Control; Progress in Systems and Control Theory; Birkhäuser: Boston, MA, USA, 1989; Volume 1. [Google Scholar]
- Hazaimeh, A. Solution of Sturm-Liouville Differential Equation via the Use of Variational Iteration Method. Master’s Thesis, Jordan University of Science and Technology, Ar-Ramtha, Jordan, May 2020. [Google Scholar]
- Adomian, G. A review of the decomposition method and some recent results for nonlinear equations. Math. Comput. Model. 1990, 13, 17–43. [Google Scholar] [CrossRef]
- Singh, N.; Kumar, M. Adomian decomposition method for computing eigen-values of singular Sturm-Liouville problems. Natl. Acad. Sci. Lett. 2013, 36, 311–318. [Google Scholar] [CrossRef]
LVIM | VIM | ||||
---|---|---|---|---|---|
0 | 0.5 | 0.5 | 0.5 | ||
0.2 | 0.409365 | 0.409365 | 0.409364 | ||
0.4 | 0.33516 | 0.33516 | 0.33511 | 0.0000497178 | |
0.6 | 0.274406 | 0.274406 | 0.274056 | 0.000350104 | |
0.8 | 0.224664 | 0.224664 | 0.223415 | 0.00124996 | |
1 | 0.18394 | 0.18394 | 0.180819 | 0.00312102 |
LVIM | AVIM | ||||
---|---|---|---|---|---|
0 | 0.5 | 0.5 | 0.5 | ||
0.2 | 0.409365 | 0.409365 | 0.409362 | ||
0.4 | 0.33516 | 0.33516 | 0.335152 | ||
0.6 | 0.274406 | 0.274406 | 0.274412 | ||
0.8 | 0.224664 | 0.224664 | 0.224658 | ||
1.0 | 0.18394 | 0.18394 | 0.184125 |
Sinc–Galerkin | |||
---|---|---|---|
0 | 0.5 | 0.5 | |
0.2 | 0.409365 | 0.409365 | |
0.4 | 0.335160 | 0.335160 | |
0.6 | 0.274406 | 0.274405 | |
0.8 | 0.224664 | 0.224665 | |
1.0 | 0.183940 | 0.183941 |
m | Sinc–Galerkin () | LVIM () |
---|---|---|
0 | ||
1 | ||
2 |
x | Sinc-Solution with () | LVIM () |
---|---|---|
0 | 0 | 0 |
0 | 0 |
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Al-Khaled, K.; Hazaimeh, A. Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems. Symmetry 2020, 12, 1179. https://doi.org/10.3390/sym12071179
Al-Khaled K, Hazaimeh A. Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems. Symmetry. 2020; 12(7):1179. https://doi.org/10.3390/sym12071179
Chicago/Turabian StyleAl-Khaled, Kamel, and Ashwaq Hazaimeh. 2020. "Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems" Symmetry 12, no. 7: 1179. https://doi.org/10.3390/sym12071179
APA StyleAl-Khaled, K., & Hazaimeh, A. (2020). Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems. Symmetry, 12(7), 1179. https://doi.org/10.3390/sym12071179