AN ALGORITHM FOR THE SOLUTION OF FUZZY FRACTIONAL DIFFERENTIAL EQUATION

Abstract

In this paper the variational iteration method is used to compute the solution for the fuzzy fractional order partial differential equations with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions.

We propose an algorithm based on $\alpha$-cut of a fuzzy set. Finally we present some examples by using

our proposed algorithm.

Keywords

• fuzzy number,the fuzzy fractional differential equation

References

1. S.Arshad, V.Lupulescu. Fractional differential equation with the fuzzy initial conditions. Elect J Differ Equat {34} {2011}1--8.
2. S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Analysis {74}, {2011} 3685--3693
3. L. S. Chadli, A. Harir and S. Melliani, Fuzzy Euler differential equation, SOP Transactions on Applied Mathematics, volume 2, number 1, January 2015.
4. L. S. Chadli, A. Harir and S. Melliani, Solutions of fuzzy heat-like equations by variational iterative method , Annals of Fuzzy Mathematics and Informatics, Volume 10, number 1, 2015, pp 29-44.
5. L. S. Chadli, A. Harir and S. Melliani, Solutions of fuzzy wave-like equations by variational iteration method. International Annals of Fuzzy Mathematics and Informatics, volume 8, number 4, 2014, 527-547 .
6. P. Diamond, P.E. Kloeden, {Metric Spaces of Fuzzy Sets, Theory and Applications,} World Scienific, Singapore, {1994}.
7. B.Dumitru, {Machado José Antnio Tenreiro}, Luo Albert CJ. Fract Dynam Control 2011.
8. R. Getschel, W.Voxman, {Elementary fuzzy calculus,} Fuzzy sets and Systems {18} {1986} 31--43.

9. Guo-cheng Wu, E.W.M. Lee,{Fractional variational iteration method and its application},Physics Letters A {374} {2010} 2506--2509
10. J.H. He, {Variational iteration method-a kind of non-linear analytical technique}: some examples, Int. J. Non-Linear Mech. {34} {1999} 699--708.
11. J.H. He, G.C. Wu, F. Austin,{The variational iteration method which should be followed},Nonl. Sci. Lett. A {1} {2010} 1--30.
12. J. H. He, {Some asymptotic methods for strongly nonlinear equations, } Int. J. Mod. Phys., B { 20} {2006} 1141--1199.
13. J.H. He, {Approximate analytical solution for seepage flow with fractional derivatives in porous media,} Comput. Methods Appl. Mech. Eng. 167 {1998} 57--68.
14. G. Jumarie,{Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,}Appl. Math. Lett. {22 }, {2009} 378--385.
15. O. Kaleva.{ Fuzzy differential equations}. {Fuzzy Sets and Systems}, {24}, {1987} 301--317.
16. M. Ma, M, Friedman, A. Kandel, {A new fuzzy arithmetic, } Fuzzy Sets and Systems {108}, {1999} 83--90.
17. Mehran Mazandarani, Ali Vahidian Kamyad, {Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem},Commun Nonlinear Sci Numer Simulat {18} {2013} 12--21
18. Meral FC, Royston TJ, Magi R.{Fractional calculus in viscoelasticity, an experimental study}. Commun Nonlinear Sci Numer Simulat {15} {2010} 939--45
19. S.Salahshour, T.Allahviranloo, S.Abbasbandy.{Solving fuzzy fractional equations by fuzzy Laplace transforms}. Commun Nonlinear Sci Numer Simul {17}{2012} 1372--1381.
20. Das Saptarshi, Pan Indranil. {Fractional order signal processing}, introductory concepts and applications. Technol Eng 2011.
21. AM. Wazwaz, {The variational iteration method for solving linear and nonlinear systems of PDEs, } Comput. Math. Appl. { 54} {2007} 95--902.