Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method
Applied and Computational Mathematics
Volume 7, Issue 2, April 2018, Pages: 58-70
Received: Feb. 4, 2018; Accepted: Feb. 24, 2018; Published: Mar. 22, 2018
Views 1179      Downloads 120
Authors
Abdrhaman Mahmoud, School of Mathematical Sciences, Dalian University of Technology, Dalian, China; Department of Mathematics, Faculty of Sciences and Technology, Omdurman Islamic University, Omdurman, Sudan
Bo Yu, School of Mathematical Sciences, Dalian University of Technology, Dalian, China
Xuping Zhang, School of Mathematical Sciences, Dalian University of Technology, Dalian, China
Article Tools
Follow on us
Abstract
In this paper, the eigenfunction expansion method (EEM) is applied to find numerical solutions for variable-coefficient fourth-order ordinary differential equations (ODEs) with polynomial nonlinearity. The symmetry of the solution set for the resulting system of polynomial equations obtained from EEM of the problem is analyzed. The symmetric homotopy method is constructed to calculate all solutions of the discretization system for the problem. Due to the exploitation of symmetry, the number of computations is reduced. Numerical examples are presented to demonstrate the efficiency of the presented homotopy method.
Keywords
Fourth-Order ODEs, System of Polynomial Equations, Homotopy Continuation Method, Numerical Algebraic Geometry, Symmetry Group
To cite this article
Abdrhaman Mahmoud, Bo Yu, Xuping Zhang, Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method, Applied and Computational Mathematics. Vol. 7, No. 2, 2018, pp. 58-70. doi: 10.11648/j.acm.20180702.14
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, Journal of Mathematical Analysis and Applications, 270 (2002), pp. 357–368.
[2]
Y. Yang, Fourth-order two-point boundary value problems, Proceedings of the American Mathematical Society, (1988), pp. 175–180.
[3]
G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, Journal of Mathematical Analysis and Applications, 343 (2008), pp. 1166–1176.
[4]
M. do Rosário Grossinho, L. Sanchez, and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Applied Mathematics Letters, 18 (2005), pp. 439–444.
[5]
L. Greenberg and M. Marletta, Numerical methods for higher order sturm-liouville problems, Journal of Computational and Applied Mathematics, 125 (2000), pp. 367–383.
[6]
Z. S. Aliyev and F. M. Namazov, Spectral properties of a fourth-order eigenvalue problem with spectral parameter in the boundary conditions, Electronic Journal of Differential Equations, 2017 (2017), pp. 1–11.
[7]
R. P. Agarwal, Boundary value problems for higher order differential equations, tech. report, 1979.
[8]
G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Non-linear Analysis: Theory, Methods &Applications, 68 (2008), pp. 3646–3656.
[9]
X. L. Liu and W. T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters, Mathematical and Computer Modelling, 46 (2007), pp. 525–534.
[10]
A. Cabada, R. Precup, L. Saavedra, and S. A. Tersian, Multiple positive solutions to a fourth-order boundary-value problem, Electronic Journal of Differential Equations, 2016 (2016), pp. 1–18.
[11]
X. Zhang, J. Zhang, and B. Yu, Eigenfunction expansion method for multiple solutions of semilinear elliptic equations with polynomial nonlinearity, SIAM Journal on Numerical Analysis, 51 (2013), pp. 2680–2699.
[12]
R. L. Burden and J. D. Faires, Numerical analysis, Cengage Learning, 2011.
[13]
J. Alexander and J. A. Yorke, The homotopy continuation method: numerically implementable topological procedures, Transactions of the American Mathematical Society, 242 (1978), pp. 271–284.
[14]
T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta numerica, 6 (1997), pp. 399 436.
[15]
T. Y. Li, Numerical solution of polynomial systems by homotopy continuation methods, Handbook of numerical analysis, 11 (2003), pp. 209–304.
[16]
J. Verschelde, Algorithm 795: Phcpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Transactions on Mathematical Software (TOMS), 25 (1999), pp. 251–276.
[17]
T. L. Lee, T. Y. Li, and C. H. Tsai, Hom4ps-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method, Computing, 83 (2008), pp. 109–133.
[18]
D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, Bertini: Software for numerical algebraic geometry (2006), Software available at http://bertini. nd. edu.
[19]
A. J. Sommese and C. W. Wampler II, The Numerical solution of systems of polynomials arising in engineering and science, World Scientific, 2005.
[20]
E. L. Allgower, D. J. Bates, A. J. Sommese, and C. W. Wampler, Solution of polynomial systems derived from differential equations, Computing, 76 (2006), pp. 1–10.
[21]
X. Zhang, J. Zhang, and B. Yu, Symmetric homotopy method for discretized elliptic equations with cubic and quantic nonlinearities, Journal of Scientific Computing, 70 (2017), pp. 1316–1335.
[22]
W. Hao, J. D. Hauenstein, B. Hu, and A. J. Sommese, A bootstrapping approach for computing multiple solutions of differential equations, Journal of Computational and Applied Mathematics, 258 (2014), pp. 181–190.
[23]
S. M. Khalkhali, S. Heidarkhani, and A. Razani, Infinitely many solutions for a fourth-order boundary-value problem, Electronic Journal of Differential Equations, 2012 (2012), pp. 1–14.
ADDRESS
Science Publishing Group
548 FASHION AVENUE
NEW YORK, NY 10018
U.S.A.
Tel: (001)347-688-8931