Solving Fractional Order Differential Problems using Fuzzy Transform

Awni Abu-Saman, Huda S. EL-Zerii


In this paper we give some background on the main concepts of the theory of fractional calculus and Grünwald formula for the fractional derivative, we also introduce Fuzzy Transform as a new technique for solving fractional differential equations. Fuzzy transform already proved itself in solving many problems in different branches, which encourage us to use it as a technique to approximate the solution of Fractional Differential Equations. The Fuzzy transform will be applied on a fractional order differential equations. The numerical algorithm will be implemented as a user-subroutine to the mathematical code MATLAB. We have introduced a numerical example of fractional order differential equation. Results are obtained for different fractional values and different partitions with triangular and Sinusoidal shaped basic functions and compared with the analytical solution.


Fractional order differential equation; Fuzzy transform; Basic functions; Numerical algorithm

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Abu-Saman A, El-Zerii H, Numerical Solution of initial value problems using Fuzzy Transform, International Journal of Applied Engineering Research, To Appear.

Achar B.N.N., Hanneken J. W., Enck T., and Clarke T., Dynamics of the fractional oscillator, Physica A 297, 2001, pp 361-367.

Chen G., Tat Pham T., Introduction to fuzzy set, fuzzy logic, and fuzzy control system, CRC, 2000.

Dankova M. and Valasek R., Full fuzzy transform and the problem of image fusion, Journal of Electrical Engineering, vol. 57, 2006, pp 82-84.

Deng Z., Singh V. P., Bengtsson L., Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng., vol. 130, 2004, pp 422-431.

Diethelm K., Freed A, Modeling and simulating viscoelastic and viscoplastic materials using nonlinear, fractional-order differential equations, submitted for publications, 2007.

Liu F., Ahn V., Turner I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math.,vol. 166, 2004, pp209-219.

Meerschaert M. M., The fractional calculus Project, MAA Student Lecture, Phoenix, January, 2004.

Patane G., Fuzzy transform and least-squares approximation, Elsevier, 2010.

Perfilieva I., Fuzzy transforms in image compressions and fusion , Acta Mathematica Universitatis Ostraviensis, vol 15, no.1, 2007, PP 27-37.

Perfilieva I., and Hodakova P., Fuzzy and Fourier Transforms, EUSFLAT-LFA, 2011, PP 452-456.

Perfilieva I., and Valasek R., Data compression on the basis of fuzzy transform, EUSFLAT-LFA conference, 2005, pp 663-668.

Perfilieva I., Novak V., Dvorak A., Fuzzy transform in the analysis of data, International Journal of Approximate Reasoning, vol 48, 2008, pp 36-46.

Perfilieva I., Fuzzy transforms and universal approximation, EUSFLAT conference, 2003, pp 529-533.

Perfilieva I., Approximating models based on fuzzy transforms, EUSFLAT-LFA conference, 2005, pp 645-650.

Podlubny I., Fractional differential equations, Academic Press, 1999.

Samko S.G., Kilbas A. A. and Marichev O.I., Fractional integrals and derivatives theory and applications, Gordon and Breach Science Publishers, 1993.

Stefanini L., Fuzzy transform and smooth functions, IFSA-EUSFLAT, 2009, PP579-584

Stepnicka M. and Valasek R., Fuzzy transform and their application to wave equation, Journal of Electrical Engineering , vol. 12, 2004, pp 7-10.

Stepnika M., Fuzzy transform and its applications to problems in engineering practice, PH.D. Thesis "Ostrava Uni.", 2007.

Tadjeran C., Meerschaert M. and Scheffler H., A second-order accurate numerical approximataion for the fractional diffusion equation, Journal of Computational Physics,vol. 213, 2006, pp 205-213.

Tadjeran C., Meerschaert M. and Scheffler H., Finite difference methods for two-dimensional fractional dispersion equation, Journal of Computational Physics, vol. 211, 2006, pp 249-261.

Zadah L., Fuzzy set theory, Information and control 8th, 1965, pp 338-353.


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