It is
well known that the matrix equations play a significant role in engineering and
applicable sciences. In this research article, a new modification of the
homotopy perturbation method (HPM) will be proposed to obtain the approximated
solution of the matrix equation in the form AX = B. Moreover, the conditions are
deduced to check the convergence of the homotopy series. Numerical
implementations are adapted to illustrate the properties of the modified
method.
References
[1]
Bartels, R. and Stewart, G. (1994) Solution of the Matrix Equation AX+XB=C. Circuits, Systems and Signal Processing, 13, 820-826.
[2]
Benner, P. (2008) Large-Scale Matrix Equations of Special Type. Numerical Linear Algebra with Applications, 15, 747-754. http://dx.doi.org/10.1002/nla.621
[3]
Golub, G., Nash, S. and Van Loan, Ch. (1979) A Hessenberg-Schur Method for the Problem AX+XB=C. IEEE Transactions on Automatic Control, 24, 909-913. http://dx.doi.org/10.1109/TAC.1979.1102170
[4]
Sadeghi, A., Ahmad, M.I., Ahmad, A. and Abbasnejad, M.E. (2013) A Note on Solving the Fuzzy Sylvester Matrix Equation. Journal of Computational Analysis and Applications, 15, 10-22.
[5]
Sadeghi, A., Abbasbandy, S. and Abbasnejad, M.E. (2011) The Common Solution of the Pair of Fuzzy Matrix Equations. World Applied Sciences, 15, 232-238.
[6]
Sadeghi, A., Abbasnejad, M.E. and Ahmad, M.I. (2011) On Solving Systems of Fuzzy Matrix Equation. Far East Journal of Applied Mathematics, 59, 31-44.
[7]
He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 57-62. http://dx.doi.org/10.1016/s0045-7825(99)00018-3
[8]
He, J.H. (2000) A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems. International Journal of Non-Linear Mechanics, 35, 37-43. http://dx.doi.org/10.1016/S0020-7462(98)00085-7
[9]
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
[10]
Keramati, B. (2007) An Approach to the Solution of Linear System of Equations by He’s Homotopy Perturbation Method. Chaos, Solitons & Fractals, 41, 152-156. http://dx.doi.org/10.1016/j.chaos.2007.11.020
[11]
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
[12]
Edalatpanah, S.A. and Rashidi, M.M. (2014) On the Application of Homotopy Perturbation Method for Solving Systems of Linear Equations. International Scholarly Research Notices, 2014, Article ID: 143512. http://dx.doi.org/10.1155/2014/143512
[13]
Saeidian, J., Babolian, E. and Aziz, A. (2015) On a Homotopy Based Method for Solving Systems of Linear Equations. TWMS Journal of Pure and Applied Mathematics, 6, 15-26.
[14]
Khani, M.H., Rashidinia, J. and Zia 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. http://dx.doi.org/10.4236/alamt.2015.53012
[15]
Laub, A.J. (2005) Matrix Analysis for Scientists and Engineers. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9780898717907
[16]
Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. 2nd Edition, SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9780898718003
