| Peer-Reviewed

Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method

Received: 4 February 2018     Accepted: 24 February 2018     Published: 22 March 2018
Views:       Downloads:
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.

Published in Applied and Computational Mathematics (Volume 7, Issue 2)
DOI 10.11648/j.acm.20180702.14
Page(s) 58-70
Creative Commons

This is an Open Access article, 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 or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Fourth-Order ODEs, System of Polynomial Equations, Homotopy Continuation Method, Numerical Algebraic Geometry, Symmetry Group

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.
Cite This Article
  • APA Style

    Abdrhaman Mahmoud, Bo Yu, Xuping Zhang. (2018). Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method. Applied and Computational Mathematics, 7(2), 58-70. https://doi.org/10.11648/j.acm.20180702.14

    Copy | Download

    ACS Style

    Abdrhaman Mahmoud; Bo Yu; Xuping Zhang. Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method. Appl. Comput. Math. 2018, 7(2), 58-70. doi: 10.11648/j.acm.20180702.14

    Copy | Download

    AMA Style

    Abdrhaman Mahmoud, Bo Yu, Xuping Zhang. Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method. Appl Comput Math. 2018;7(2):58-70. doi: 10.11648/j.acm.20180702.14

    Copy | Download

  • @article{10.11648/j.acm.20180702.14,
      author = {Abdrhaman Mahmoud and Bo Yu and Xuping Zhang},
      title = {Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method},
      journal = {Applied and Computational Mathematics},
      volume = {7},
      number = {2},
      pages = {58-70},
      doi = {10.11648/j.acm.20180702.14},
      url = {https://doi.org/10.11648/j.acm.20180702.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180702.14},
      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.},
     year = {2018}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method
    AU  - Abdrhaman Mahmoud
    AU  - Bo Yu
    AU  - Xuping Zhang
    Y1  - 2018/03/22
    PY  - 2018
    N1  - https://doi.org/10.11648/j.acm.20180702.14
    DO  - 10.11648/j.acm.20180702.14
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 58
    EP  - 70
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20180702.14
    AB  - 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.
    VL  - 7
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • School of Mathematical Sciences, Dalian University of Technology, Dalian, China

  • School of Mathematical Sciences, Dalian University of Technology, Dalian, China

  • School of Mathematical Sciences, Dalian University of Technology, Dalian, China

  • Sections