Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations
Most real world systems express themselves formally
as a set of nonlinear algebraic equations. As applications grow, the
size and complexity of these equations also increase. In this work, we
highlight the key concepts in using the homotopy analysis method
as a methodology used to construct efficient iteration formulas for
nonlinear equations solving. The proposed method is experimentally
characterized according to a set of determined parameters which
affect the systems. The experimental results show the potential and
limitations of the new method and imply directions for future work.
[1] S. Abbasbandy, Y. Tan, S.J. Liao, Newton-homotopy analysis method for
nonlinear equations, Appl. Math. Comput. 188(2007) 1794-1800.
[2] F. Awawdeh, On New Iterative Method for Solving Systems of Nonlinear
Equations, Numer. Algorithms. 54(2010) 395-409.
[3] F. Awawdeh, M. Khandaqji, Z. Mustafa, A new approach for the solution
of the electrostatic potential differential equations, Mathematical
Problems in Engineering, 2009 (2009) 1-11.
[4] F. Awawdeh, H.M. Jaradat, O. Alsayyed, Solving System of DAEs by
Homotopy Analysis Method, Chaos, Solitons and Fractals, 42(2009)
1422-1427.
[5] F. Awawdeh, A. Adawi, Z. Mustafa, Solutions of the SIR Models of
Epidemics Using HAM, Chaos, Solitons and Fractals, 42(2009) 3047-
3052.
[6] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[7] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl.
Math. Comput. 47(2004) 499-513.
[8] S.J. Liao, Notes on the homotopy analysis method: Some definitions and
theorems, Commun Nonlinear Sci Numer Simul. 14(2009) 983-997.
[9] S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear
differential equations, Commun Nonlinear Sci Numer Simul. 15(2010)
2003-2016.
[1] S. Abbasbandy, Y. Tan, S.J. Liao, Newton-homotopy analysis method for
nonlinear equations, Appl. Math. Comput. 188(2007) 1794-1800.
[2] F. Awawdeh, On New Iterative Method for Solving Systems of Nonlinear
Equations, Numer. Algorithms. 54(2010) 395-409.
[3] F. Awawdeh, M. Khandaqji, Z. Mustafa, A new approach for the solution
of the electrostatic potential differential equations, Mathematical
Problems in Engineering, 2009 (2009) 1-11.
[4] F. Awawdeh, H.M. Jaradat, O. Alsayyed, Solving System of DAEs by
Homotopy Analysis Method, Chaos, Solitons and Fractals, 42(2009)
1422-1427.
[5] F. Awawdeh, A. Adawi, Z. Mustafa, Solutions of the SIR Models of
Epidemics Using HAM, Chaos, Solitons and Fractals, 42(2009) 3047-
3052.
[6] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[7] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl.
Math. Comput. 47(2004) 499-513.
[8] S.J. Liao, Notes on the homotopy analysis method: Some definitions and
theorems, Commun Nonlinear Sci Numer Simul. 14(2009) 983-997.
[9] S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear
differential equations, Commun Nonlinear Sci Numer Simul. 15(2010)
2003-2016.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50440", author = "Rafat Alshorman and Safwan Al-Shara' and I. Obeidat", title = "Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations", abstract = "Most real world systems express themselves formally
as a set of nonlinear algebraic equations. As applications grow, the
size and complexity of these equations also increase. In this work, we
highlight the key concepts in using the homotopy analysis method
as a methodology used to construct efficient iteration formulas for
nonlinear equations solving. The proposed method is experimentally
characterized according to a set of determined parameters which
affect the systems. The experimental results show the potential and
limitations of the new method and imply directions for future work.", keywords = "Nonlinear Algebraic Equations, Iterative Methods, Homotopy
Analysis Method.", volume = "7", number = "3", pages = "330-3", }