We present an efficient numerical algorithm for solution of the fuzzy linear systems (FLS) based on He’s homotopy perturbation method (HPM). Moreover, the convergence properties of the proposed method have been analyzed and also comparisons are made between Adomian’s decomposition method (ADM) and the proposed method. The results reveal that our method is effective and simple. 1. Introduction Consider the following linear system: where denotes a vector in a finite-dimensional space and . Linear systems are of fundamental importance in various fields of science and engineering and there are numerous methods to find a solution for them; see [1–8] and the references therein. When coefficients of the system are imprecise, accessing the solution with no ambiguity will be difficult. Therefore, fuzzy logic was proposed by Zadeh (1965) and encountered seriously over Aristotle classic logic. Professor Zadeh discussed fuzzy set theory, fuzzy number concept, and arithmetical operation with these numbers in [9–12] and following that many articles and books were published in fuzzy systems. We refer the reader to [13] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [14], fuzzy differential equations [15], particle physics [16, 17], game theory [18], optimization [19], and especially fuzzy linear systems [20–24]. Fuzzy number arithmetic is widely applied and useful in computation of linear system whose parameters are all or partially represented by fuzzy numbers. Friedman et al. [20] introduced a general model for solving a fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy linear system by a crisp linear system. Furthermore, some iterative methods have been presented for solving such fuzzy linear systems [25–32]. In this paper, we solve fuzzy linear systems (FLS) via an analytical method called homotopy perturbation method (HPM). This method was first proposed by the Chinese mathematician He in 1999 [33] and was further developed and improved by him [34–36]. This author presented a homotopy perturbation technique based on the introduction of homotopy in topology coupled with the traditional perturbation method for the solution of algebraic equations. This technique provides a summation of an infinite series with easily computable terms, which converges rapidly to the solution of the problem. In the literature, various authors
References
[1]
R. S. Varga, Matrix Iterative Analysis, Springer, Berlin, Germany, 2nd edition, 2000.
[2]
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
[3]
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2003.
[4]
H. S. Najafi and S. A. Edalatpanah, “On the convergence regions of generalized AOR methods for linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 156, pp. 859–866, 2013.
[5]
H. S. Najafi and S. A. Edalatpanah, “A new modified SSOR ieration method for solving augmented linear systems,” International Journal of Computer Mathematics, 2013.
[6]
H. S. Najafi and S. A. Edalatpanah, “Comparison analysis for improving preconditioned SOR-type iterative method,” Numerical Analysis and Applications, vol. 6, pp. 62–70, 2013.
[7]
H. S. Najafi and S. A. Edalatpanah, “Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems,” RAIRO, vol. 47, pp. 59–71, 2013.
[8]
H. S. Najafi and S. A. Edalatpanad, “On application of Liao’s method for system of linear equations,” Ain Shams Engineering Journal, vol. 4, pp. 501–505, 2013.
[9]
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
[10]
L. A. Zadeh, “A fuzzy-set-theoretic interpretation of linguistic hedges,” Journal of Cybernetics, vol. 2, no. 3, pp. 4–34, 1972.
[11]
L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975.
[12]
L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 3–28, 1978.
[13]
A. Kaufmann and M. M. Gupta, Introduction Fuzzy Arithmetic, Van Nostrand Reinhold, New York, NY, USA, 1985.
[14]
J. H. Park, “Intuitionistic fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1039–1046, 2004.
[15]
M. Friedman, M. Ming, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35–48, 1999.
[16]
M. S. Elnaschie, “A review of E infinity theory and the mass spectrum of high energy particle physics,” Chaos, Solitons and Fractals, vol. 19, no. 1, pp. 209–236, 2004.
[17]
M. S. Elnaschie, “The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 495–511, 2004.
[18]
H. S. Najafi and S. A. Edalatpanad, “On the Nash equilibrium solution of fuzzy bimatrix games,” International Journal of Fuzzy Systems and Rough Systems, vol. 5, no. 2, pp. 93–97, 2012.
[19]
H. S. Najafi and S. A. Edalatpanad, “A note on ‘a new method for solving fully fuzzy linear programming problems’,” Applied Mathematical Modelling, vol. 37, pp. 7865–7867, 2013.
[20]
M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201–209, 1998.
[21]
D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, NewYork, NY, USA, 1980.
[22]
D. Dubois and H. Prade, “Systems of linear fuzzy constraints,” Fuzzy Sets and Systems, vol. 3, no. 1, pp. 37–48, 1980.
[23]
J. J. Buckley and Y. Qu, “Solving systems of linear fuzzy equations,” Fuzzy Sets and Systems, vol. 43, no. 1, pp. 33–43, 1991.
[24]
M. Friedman, M. Ming, and A. Kandel, “Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, pp. 55–58, 2000.
[25]
T. Allahviranloo, “Numerical methods for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 493–502, 2004.
[26]
T. Allahviranloo, “The Adomian decomposition method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 553–563, 2005.
[27]
M. Dehghan and B. Hashemi, “Iterative solution of fuzzy linear systems,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 645–674, 2006.
[28]
K. Wang and B. Zheng, “Symmetric successive overrelaxation methods for fuzzy linear systems,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 891–901, 2006.
[29]
B. Asady and P. Mansouri, “Numerical solution of fuzzy linear system,” International Journal of Computer Mathematics, vol. 86, no. 1, pp. 151–162, 2009.
[30]
H. S. Najafi and S. A. Edalatpanah, “The block AOR iterative methods for solving fuzzy linear systems,” The Journal of Mathematics and Computer Science, vol. 4, pp. 527–535, 2012.
[31]
H. S. Najafi and S. A. Edalatpanah, “Preconditioning strategy to solve fuzzy linear systems (FLS),” International Review of Fuzzy Mathematics, vol. 7, no. 2, pp. 65–80, 2012.
[32]
H. S. Najafi and S. A. Edalatpanad, “An improved model for iterative algorithms in fuzzy linear systems,” Computational Mathematics and Modeling, vol. 24, pp. 443–451, 2013.
[33]
J. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
[34]
J. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
[35]
J. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.
[36]
J. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
[37]
A. Beléndez, A. Hernández, T. Beléndez, E. Fernández, M. L. Alvarez, and C. Neipp, “Application of He’s homotopy perturbation method to the Duffing-harmonic oscillator,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 79–88, 2007.
[38]
T. ？zi？ and A. Yildirim, “A note on He's homotopy perturbation method for van der Pol oscillator with very strong nonlinearity,” Chaos, Solitons and Fractals, vol. 37, no. 3, pp. 989–991, 2007.
[39]
J. I. Ramos, “Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 400–408, 2008.
[40]
L. Cveticanin, “Application of homotopy-perturbation to non-linear partial differential equations,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 221–228, 2009.
[41]
B. Keramati, “An approach to the solution of linear system of equations by He's homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 152–156, 2009.
[42]
M. Mehrabinezhad and J. Saberi-Nadjafi, “Application of He's homotopy perturbation method to linear programming problems,” International Journal of Computer Mathematics, vol. 88, no. 2, pp. 341–347, 2011.
[43]
A. Frommer and D. B. Szyld, “H-splittings and two-stage iterative methods,” Numerische Mathematik, vol. 63, no. 1, pp. 345–356, 1992.
[44]
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, The Netherlands, 1994.
[45]
G. Adomian and R. Rach, “On the solution of algebraic equations by the decomposition method,” Journal of Mathematical Analysis and Applications, vol. 105, no. 1, pp. 141–166, 1985.
