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

Abdrhaman Mahmoud; Bo Yu; Xuping Zhang

doi:doi:10.11648/j.acm.20180702.14

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

Abdrhaman Mahmoud, Bo Yu, Xuping Zhang

Published in Applied and Computational Mathematics (Volume 7, Issue 2)

Received: 4 February 2018 Accepted: 24 February 2018 Published: 22 March 2018

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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), 2024. Published by Science Publishing Group

Keywords

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

References

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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

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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

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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

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@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}
}

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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  -

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Author Information

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

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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

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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

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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