Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations

Authors: Rafat Alshorman, Safwan Al-Shara', I. Obeidat

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.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1054944

References:

[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.