Application of Different H(x) in Homotopy Analysis Methods for Solving Systems of Linear Equations

Mohammad Hasan Khani; Jalil Rashidinia; Sajjad Zia Borujeni

doi:10.4236/alamt.2015.53012

Home > Journals > Physics & Mathematics > ALAMT

Advances in Linear Algebra & Matrix Theo... > Vol.5 No.3, September 2015

Application of Different H(x) in Homotopy Analysis Methods for Solving Systems of Linear Equations ()

Mohammad Hasan Khani¹, Jalil Rashidinia², Sajjad Zia Borujeni^1,2*

¹Department of Mathematics, Islamic Azad University, Shahin Shahr Branch, Isfahan, Iran.
²Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.

DOI: 10.4236/alamt.2015.53012 PDF HTML XML 3,734 Downloads 4,347 Views Citations

Abstract

In this paper, we present homotopy analysis method (HAM) for solving system of linear equations and use of different H(x) in this method. The numerical results indicate that this method performs better than the homotopy perturbation method (HPM) for solving linear systems.

Keywords

HAM, HPM, Linear System

Share and Cite:

Khani, M. , Rashidinia, J. and Borujeni, S. (2015) Application of Different H(x) in Homotopy Analysis Methods for Solving Systems of Linear Equations. Advances in Linear Algebra & Matrix Theory, 5, 129-137. doi: 10.4236/alamt.2015.53012.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Nazari, A.M. and Zia Borujeni, S. (2012) A Modified Precondition in the Gauss-Seidel Method. Advances in Linear Algebra & Matrix Theory, 1, 31-37. http://dx.doi.org/10.4236/alamt.2012.23005
[2]	Keramati, B. (2009) An Approach to the Solution of Linear System of Equations by He’s Homotopy Perturbation Method. Chaos, Solitons and Fractals, 41, 152-156. http://dx.doi.org/10.1016/j.chaos.2007.11.020
[3]	Liu, H.K. (2011) Application of Homotopy Perturbation Methods for Solving Systems of Linear Equations. Applied Mathematics and Computation, 217, 5259-5264. http://dx.doi.org/10.1016/j.amc.2010.11.024
[4]	Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai.
[5]	Morimoto, M., Harada, K., Sakakihara, M. and Sawami, H. (2004) The Gauss-Seidel Iterative Method with the Preconditioning Matrix.(I+S+S_m). Japan Journal of Industrial and Applied Mathematics, 21, 25-34. http://dx.doi.org/10.1007/BF03167430
[6]	Niki, H., Kohno, T. and Morimoto, M. (2008) The Preconditioned Gauss-Seidel Method Faster than the SOR Method. Journal of Computational and Applied Mathematics, 218, 59-71. http://dx.doi.org/10.1016/j.cam.2007.07.002
[7]	Niki, H., Kohno, T. and Abe, K. (2009) An Extended GS Method for Dense Linear System. Journal of Computational and Applied Mathematics, 231, 177-186. http://dx.doi.org/10.1016/j.cam.2009.02.005
[8]	Yusefoglu, E. (2009) An Improvement to Homotopy Perturbation Method for Solving System of Linear Equations. Computers & Mathematics with Applications, 58, 2231-2235. http://dx.doi.org/10.1016/j.camwa.2009.03.010
[9]	Yuan, J.Y. and Zontini, D.D. (2012) Comparison Theorems of Preconditioned Gauss-Seidel Methods for M-Matrices. Applied Mathematics and Computation, 219, 1947-1957. http://dx.doi.org/10.1016/j.amc.2012.08.037
[10]	Gunawardena, A.D., Jain, S.K. and Snyder, L. (1991) Modified Iterative Method for Consistent Linear Systems. Linear Algebra and Its Applications, 154-156, 123-143. http://dx.doi.org/10.1016/0024-3795(91)90376-8
[11]	Kohno, T. and Niki, H. (2009) A Note on the Preconditioner Pm=(I+Sm). Journal of Computational and Applied Mathematics, 225, 316-319. http://dx.doi.org/10.1016/j.cam.2008.07.042
[12]	He, J.H. (2003) Homotopy Perturbation Method: A New Non-Linear Analytical Technique. Applied Mathematics and Computation, 135, 73-79. http://dx.doi.org/10.1016/S0096-3003(01)00312-5
[13]	Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. SIAM Press, PHiladelphia. http://dx.doi.org/10.1137/1.9780898718003
[14]	Liao, S.J. (2003) Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton. http://dx.doi.org/10.1201/9780203491164
[15]	Liao, S.J. (2009) Notes on the Homotopy Analysis Method: Some Definitions and Theorems. Communications in Nonlinear Science and Numerical Simulation, 14, 983-997. http://dx.doi.org/10.1016/j.cnsns.2008.04.013
[16]	Sen, S. (1983) Topology and Geometry for Physicists. Academic Press, Waltham.