Application of homotopy perturbation method for fuzzy linear systems and comparison with Adomian’s decomposition method. (English) Zbl 1301.65023
Summary: We present an efficient numerical algorithm for solution of the fuzzy linear systems based on He’s homotopy perturbation method. Moreover, the convergence properties of the proposed method have been analyzed and also comparisons are made between Adomian’s decomposition method and the proposed method. The results reveal that our method is effective and simple.
Keywords:
algorithm; fuzzy linear system; He’s homotopy perturbation method; convergence; Adomian’s decomposition methodReferences:
| [1] | R. S. Varga, Matrix Iterative Analysis, Springer, Berlin, Germany, 2nd edition, 2000. · Zbl 0998.65505 |
| [2] | D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971. · Zbl 0231.65034 |
| [3] | Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2003. · Zbl 1031.65046 |
| [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. · Zbl 1262.90176 |
| [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. · Zbl 1316.65042 · doi:10.1080/00207160. 2013.792923 |
| [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. · Zbl 1299.65054 |
| [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. · Zbl 1276.90075 |
| [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. · Zbl 0139.24606 |
| [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. · Zbl 0397.68071 · doi:10.1016/0020-0255(75)90036-5 |
| [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. · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5 |
| [13] | A. Kaufmann and M. M. Gupta, Introduction Fuzzy Arithmetic, Van Nostrand Reinhold, New York, NY, USA, 1985. · Zbl 0588.94023 |
| [14] | J. H. Park, “Intuitionistic fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1039-1046, 2004. · Zbl 1060.54010 · doi:10.1016/j.chaos.2004.02.051 |
| [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. · Zbl 0931.65076 · doi:10.1016/S0165-0114(98)00355-8 |
| [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. · Zbl 1071.81501 · doi:10.1016/S0960-0779(03)00278-9 |
| [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. · Zbl 1063.81582 · doi:10.1016/j.chaos.2004.02.028 |
| [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. · Zbl 1426.90269 |
| [20] | M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201-209, 1998. · Zbl 0929.15004 · doi:10.1016/S0165-0114(96)00270-9 |
| [21] | D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, NewYork, NY, USA, 1980. · Zbl 0444.94049 |
| [22] | D. Dubois and H. Prade, “Systems of linear fuzzy constraints,” Fuzzy Sets and Systems, vol. 3, no. 1, pp. 37-48, 1980. · Zbl 0425.94029 · doi:10.1016/0165-0114(80)90004-4 |
| [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. · Zbl 0741.65023 · doi:10.1016/0165-0114(91)90019-M |
| [24] | M. Friedman, M. Ming, and A. Kandel, “Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, pp. 55-58, 2000. · Zbl 0945.15002 · doi:10.1016/S0165-0114(98)00102-X |
| [25] | T. Allahviranloo, “Numerical methods for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 493-502, 2004. · Zbl 1067.65040 · doi:10.1016/S0096-3003(03)00793-8 |
| [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. · Zbl 1069.65025 · doi:10.1016/j.amc.2004.02.020 |
| [27] | M. Dehghan and B. Hashemi, “Iterative solution of fuzzy linear systems,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 645-674, 2006. · Zbl 1137.65336 · doi:10.1016/j.amc.2005.07.033 |
| [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. · Zbl 1101.65036 · doi:10.1016/j.amc.2005.08.005 |
| [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. · Zbl 1175.65041 · doi:10.1080/00207160701621206 |
| [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. · Zbl 1328.65081 |
| [33] | J. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 |
| [34] | J. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73-79, 2003. · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5 |
| [35] | J. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006 |
| [36] | J. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005 |
| [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. Özi\cs 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. · doi:10.1016/j.chaos.2006.04.013 |
| [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. · Zbl 1146.34300 · doi:10.1016/j.chaos.2006.11.018 |
| [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. · Zbl 1197.65200 · doi:10.1016/j.chaos.2007.07.053 |
| [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. · Zbl 1198.65056 · doi:10.1016/j.chaos.2007.11.020 |
| [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. · Zbl 1211.65074 · doi:10.1080/00207160903443763 |
| [43] | A. Frommer and D. B. Szyld, “H-splittings and two-stage iterative methods,” Numerische Mathematik, vol. 63, no. 1, pp. 345-356, 1992. · Zbl 0764.65018 · doi:10.1007/BF01385865 |
| [44] | G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, The Netherlands, 1994. · Zbl 0802.65122 |
| [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. · Zbl 0552.60060 · doi:10.1016/0022-247X(85)90102-7 |
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