American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 619-627. doi: 10.3934/proc.2013.2013.619

Fuzzy system of linear equations

Purnima Pandit 1,

1. 

Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara, Gujarat, India

Received  September 2012 Revised  March 2013 Published  November 2013

Real life applications arising in various fields of Engineering and Sciences like Electrical, Civil, Economics, Dietary etc. can be modeled using system of linear equations. In such models it may happen that the values of the parameters are not known or they cannot be stated precisely only their estimation due to experimental data or experts knowledge is available. In such situation it is convenient to represent such parameters by fuzzy numbers (refer [22]). Klir, [15] gave the results for the existence of solution of linear algebraic equation involving fuzzy numbers. The method to obtain solution of system of linear equations with all the involved parameters being fuzzy is proposed here. The $\alpha$-cut technique is well known in obtaining weak solutions, (refer [7]) for fully fuzzy systems of linear equations (FFSL). In this paper, the conditions for the existence and uniqueness of the fuzzy solution are proved.
Keywords: level cut, $\alpha$-cut, Fuzzy numbers, fully fuzzy linear equations, parallel computing., weak solutions.
Mathematics Subject Classification: Primary: 03E7.
Citation: Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619
References:
[1]

S. Abbasbandy S., A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric positive-definite system of linear equations, Applied Mathematics and Computation, 171 (2005), 1184-1191.

[2]

S. Abbasbandy S., R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system of linear equations, Applied Mathematics and Computation 172 (2006), 633-643.

[3]

S. Abbasbandy and A. Jafarian, Steepest descent method for system of fuzzy linear equations, Applied Mathematics and Computation 175 (2006), 823-833 .

[4]

T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation 155 (2004), 493-502.

[5]

T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation 162 (2004), 189-196.

[6]

T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation 163 (2005), 553-563.

[7]

T. Allahviranloo, M. Ghanbari, A. A. Hosseinzadeh, E. Haghi and R. Nuraei, A note on fuzzy linear systems, Fuzzy Sets and Systems 177(1) (2011), 87-92.

[8]

T. Allahviranloo, E. Ahmady and N. Ahmady, A method for solving nth order fuzzy linear differential equations, International Journal of Computer Mathematics, 86(4) (2009), 730-742.

[9]

D. Behera and S. Chakraverty, Solution of Fuzzy System of Linear Equations with Polynomial Parameteric form, Applications and Applied Mathematics, 7(2) (2012), 648-657.

[10]

J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems, 38 (1990), 43-49.

[11]

M. Dehghan, B. Hashemi and M. Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics and Computation, 179 (2006), 328-343.

[12]

M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear system using the decomposition procedure, Applied Mathematics and Computation, 182 (2006), 1568-1580.

[13]

D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press,New York, 1980.

[14]

M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 (1998), 201-209.

[15]

G. Klir and B.Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice Hall, 1997.

[16]

M. Matinfar, S. H. Nasseri and M. Sohrabi, Solving Fuzzy Linear System of Equations by Using Householder Decomposition Method, Applied Mathematical Sciences, 2(52) (2008), 2569-2575.

[17]

S. H. Nasseri, M. Abdi and B. Khabiri, An Application of Fuzzy linear System of Equations in Economic Sciences, Australian Journal of Basic and Applied Sciences, 5(7) (2011), 7-14.

[18]

S. H. Nasseri and M. Sohrabi, Gram-Schmidt approach for linear System of Equations with fuzzy parameters, The Journal of Mathematics and Computer Science, 1(2) (2010), 80-89.

[19]

M. J. Quinn, Parallel Computing Theory and Practice, Oregon State University, Second Edition. (2002).

[20]

T. Rahgooy, H. Sadoghi and R. Monsefi, Fuzzy Complex System of linear equations Applied to Circuit Analysis, International Journal of Computer and Electrical Engineering, 1(5) (2009), 535-541.

[21]

A. Sadeghi, I. M. Ahmad and A. F. Jameel, Solving Systems of Fuzzy Differential Equation, International Mathematical Forum, 6 (42) (2011), 2087-2100.

[22]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353 .

show all references

References:
[1]

S. Abbasbandy S., A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric positive-definite system of linear equations, Applied Mathematics and Computation, 171 (2005), 1184-1191.

[2]

S. Abbasbandy S., R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system of linear equations, Applied Mathematics and Computation 172 (2006), 633-643.

[3]

S. Abbasbandy and A. Jafarian, Steepest descent method for system of fuzzy linear equations, Applied Mathematics and Computation 175 (2006), 823-833 .

[4]

T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation 155 (2004), 493-502.

[5]

T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation 162 (2004), 189-196.

[6]

T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation 163 (2005), 553-563.

[7]

T. Allahviranloo, M. Ghanbari, A. A. Hosseinzadeh, E. Haghi and R. Nuraei, A note on fuzzy linear systems, Fuzzy Sets and Systems 177(1) (2011), 87-92.

[8]

T. Allahviranloo, E. Ahmady and N. Ahmady, A method for solving nth order fuzzy linear differential equations, International Journal of Computer Mathematics, 86(4) (2009), 730-742.

[9]

D. Behera and S. Chakraverty, Solution of Fuzzy System of Linear Equations with Polynomial Parameteric form, Applications and Applied Mathematics, 7(2) (2012), 648-657.

[10]

J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems, 38 (1990), 43-49.

[11]

M. Dehghan, B. Hashemi and M. Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics and Computation, 179 (2006), 328-343.

[12]

M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear system using the decomposition procedure, Applied Mathematics and Computation, 182 (2006), 1568-1580.

[13]

D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press,New York, 1980.

[14]

M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 (1998), 201-209.

[15]

G. Klir and B.Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice Hall, 1997.

[16]

M. Matinfar, S. H. Nasseri and M. Sohrabi, Solving Fuzzy Linear System of Equations by Using Householder Decomposition Method, Applied Mathematical Sciences, 2(52) (2008), 2569-2575.

[17]

S. H. Nasseri, M. Abdi and B. Khabiri, An Application of Fuzzy linear System of Equations in Economic Sciences, Australian Journal of Basic and Applied Sciences, 5(7) (2011), 7-14.

[18]

S. H. Nasseri and M. Sohrabi, Gram-Schmidt approach for linear System of Equations with fuzzy parameters, The Journal of Mathematics and Computer Science, 1(2) (2010), 80-89.

[19]

M. J. Quinn, Parallel Computing Theory and Practice, Oregon State University, Second Edition. (2002).

[20]

T. Rahgooy, H. Sadoghi and R. Monsefi, Fuzzy Complex System of linear equations Applied to Circuit Analysis, International Journal of Computer and Electrical Engineering, 1(5) (2009), 535-541.

[21]

A. Sadeghi, I. M. Ahmad and A. F. Jameel, Solving Systems of Fuzzy Differential Equation, International Mathematical Forum, 6 (42) (2011), 2087-2100.

[22]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353 .

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