Convergence of a semi-analytical method on the fuzzy linear systems

Document Type: Research Paper


1 Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran

2 Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran


In this paper, we apply the  homotopy analysis method (HAM) for solving fuzzy  linear systems and present  the necessary and sufficient conditions for the convergence of series solution obtained via the HAM. Also, we present a new criterion for choosing a proper value of convergence-control parameter $\hbar$ when the HAM is applied to linear system of equations. Comparisons are made between the results of the HAM and several well-known numerical algorithms such as Jacobi method (JM), Gauss-Seidel method (GSM), successive over relaxation method (SOR), Adomian decomposition method (ADM) and homotopy perturbation method (HPM).


bibitem{ABB3}  S. Abbasbandy, {it The application of homotopy analysis method to nonlinear equations arising in heat transfer}, Phys. Lett. A, textbf{360} (2006), 109--113.

bibitem{ALL1}  T. Allahviranloo, {it Numerical methods for fuzzy system of linear equations}, Applied Mathematics and Computation,  textbf{155} (2004), 493--502.

bibitem{ALL2} T. Allahviranloo, {it Successive over relaxation iterative  method for fuzzy system of linear
                        equations}, Applied Mathematics and Computation,  textbf{162} (2005), 189--196.

bibitem{ALL3} T. Allahviranloo, {it The Adomian decomposition method for fuzzy system of linear equations}, Applied Mathematics and Computation,  textbf{163} (2005), 553--563.

bibitem{ALL4} T. Allahviranloo and M. Ghanbari, {it Solving fuzzy linear systems by homotopy perturbation method}, International Journal of Computational Cognition,   textbf{8(2)} (2010), 24--30.

bibitem{ALL7} T. Allahviranloo, M. Ghanbari, E. Haghi, A. Hosseinzadeh and R. Nouraei, {it A note on ``Fuzzy linear systems''''''''}, Fuzzy Sets and Systems,  textbf{177(1)} (2011), 87--92.

bibitem{ALL10} T. Allahviranloo and M. Ghanbari, {it On the algebraic solution of fuzzy linear systems based on interval theory}, Applied Mathematical Modelling, textbf{36} (2012), 5360--5379.

bibitem{ALL11} T. Allahviranloo and M. Ghanbari, {it A new approach to obtain algebraic
                solution of interval linear systems}, Soft Computing,  textbf{16} (2012), 121--133.

bibitem{ALL9}  T. Allahviranloo, E. Haghi and M. Ghanbari, {it The nearest symmetric fuzzy solution for a   symmetric fuzzy linear system}, An. St. Univ. Ovidius Constanta, textbf{20(1)} (2012), 151--172.

bibitem{ALL5} T. Allahviranloo and S. Salahshour, {it Fuzzy symmetric solution of fuzzy linear systems},
                       Journal of Computational and Applied Mathematics, textbf{235(16)} (2011), 4545--4553.

bibitem{ALL8} T. Allahviranloo, R. Nuraei, M. Ghanbari, E. Haghi and A. A. Hosseinzadeh, {it A new metric for $L$-$R$ fuzzy numbers and its application in fuzzy linear systems}, Soft Computing, textbf{16} (2012), 1743-–1754.

bibitem{BAT1}  A. S. Bataineh, M. S. M. Noorani and  I. Hashim, {it Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method}, Phys. Lett. A, textbf{371} (2007) 72--82.

bibitem{BUK1}  J. J. Buckley, {it Solving fuzzy equations in economics and finance}, Fuzzy Sets and Systems, textbf{48} (1992), 289--296.

bibitem{BUK2} J. J. Buckley, {it Solving fuzzy equations}, Fuzzy Sets and Systems, textbf{50} (1992), 1--14.

bibitem{BUK3}  J. J. Buckley and Y. Qu, {it Solving systems of linear fuzzy equations}, Fuzzy Sets and Systems,textbf{43} (1991) 33--43.

bibitem{CON1}  W. Cong-Xin and M. Ming, {it Embedding problem of fuzzy number space: Part I}, Fuzzy Sets and Systems,  textbf{44} (1991), 33--38.

%bibitem{DEM1}  R. DeMarr, {it Nonnegative matrices with nonnegative inverses}, Proc. Amer. Math. Soc. (1972), %307--308.

bibitem{MEH1}  M. Dehghan and B. Hashemi, {it Iterative solution of fuzzy linear systems}, Applied Mathematics and Computation,  textbf{175} (2006), 645--674.

bibitem{FRI1}  M. Friedman, M. Ming and A. Kandel, {it Fuzzy linear systems}, Fuzzy Sets and Systems,  textbf{96}  (1998), 201--209.

bibitem{HAY1}  T. Hayat and M. Sajid, {it On analytic solution for thin film flow of a fourth grade fluid down a  vertical cylinder}, Phys. Lett. A,  textbf{361} (2007), 316--322.

bibitem{HE1}  J. H. He, {it Homotopy perturbation technique}, Comput. Methods Appl. Mech. Eng., textbf{178} (1999), 257--262.

bibitem{HE2}  J. H. He, {it Homotopy perturbation method: a new non-linear analytical technique}, Applied Mathematics and Computation,  textbf{135(1)} (2003), 73--79.

bibitem{HE3}  J. H. He, {it Application of homotopy perturbation method to non-linear wave equations}, Chaos, soitons and Fractals,  textbf{26(3)} (2005), 695--700.

bibitem{JAF}  H. Jafari, M. Saeidy and J. Vahidi,  {it The Homotopy analysis method for solving fuzzy system of linear equations}, International Journal of Fuzzy Systems, textbf{11(4)} (2009), 208--313.

bibitem{KER1}  B. Keramati, {it An approach to the solution of linear system of equations by He''''s homotopy perturbation method}, Chaos, Solitons and Fractals,  textbf{41} (2009), 152--156.

bibitem{LIA3}  S. J. Liao, {it The proposed homotopy analysis technique for the solution of nonlinear problems}, PhD thesis, Shanghai Jiao Tong University, 1992.

bibitem{LIA4}  S. J. Liao, {it An explicit totally analytic approximation of Blasius viscous flow problems}, Int. J. Non-Linear Mech., textbf{34(4)} (1999), 759--78.

bibitem{LIA1}  S. J. Liao, {it Beyond perturbation: introduction to the homotopy analysis method}, Boca Raton: Chapman and Hall, CRC Press, 2003.

bibitem{LIA6}  S. J. Liao, {it A general approach to obtain series solutions of nonlinear                differential equations}, Stud. Appl. Math.,  textbf{119} (2007), 297--355.

bibitem{LIA2} S. J. Liao, {it Notes on the homotopy analysis method: Some definitions and               theorems}, Commun. Nonlinear Sci. Numer. Simulat., textbf{14} (2009), 983--997.

bibitem{NUR}  R. Nuraei, T. Allahviranloo and  M. Ghanbari, {it Finding an inner estimation of the solution set of a fuzzy linear system}, Applied Mathematical Modelling, textbf{37} (2013), 5148--5161.

bibitem{ODI}  Z. Odibat, {it A study on the convergence of homotopy analysis method}, Applied Mathematics and Computation, textbf{217(2)} (2010), 782--789.

bibitem{SAJ1}  M. Sajid and T. Hayat, {it Comparison of HAM and HPM methods in nonlinear heat conduction and convection  equations}, Nonlinear Anal. (B), textbf{9} (2008), 2290--2295.

bibitem{SON1} L. Song and H. Zhang, {it Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation}, Phys. Lett. A,  textbf{367} (2007), 88--94.

bibitem{VAN1}  R. A Van Gorder and K. Vajravelu, {it Analytic and numerical solutions to the Lane-Emden equation}, Phys.  Lett. A,  textbf{372} (2008), 6060--6065.

bibitem{VAN2}  R. A. Van Gorder and K. Vajravelu, {it On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: a general approach}, Commun. Nonlinear Sci. Numer. Simulat., textbf{14} (2009), 4078--4089.