A novel approach of variable order derivative Theory and Methods

Volume 9, Issue 6, pp 4867--4876 http://dx.doi.org/10.22436/jnsa.009.06.122

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Authors

Badr Saad T. Alkahtani - Department of mathematics, colle of science, King Saud University, P. O. Box 1142, Riyadh, 11989, Saudi Arabia. Ilknur Koca - Department of Mathematics, Faculty of Sciences, Mehmet Akif Ersoy University, 15100, Burdur, Turkey. Abdon Atangana - Institute for groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300 Bloemfontein, South Africa.

Abstract

In order to solve problems posed while using the concept of fractional variable order derivative, we introduce in this work a novel fractional variable order derivative. Our derivative has no singular kernel, this allows it to well-describe the effect of memory. We present the relationship between the new derivative with the well-known integral transforms. We present exact solution of some basic associated differential equations. We presented the numerical approximation of the derivative for first and second order approximation.

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Keywords

• New fractional variable order derivative
• properties
• numerical method.

MSC

•  26A33
•  35B35
•  34A12

References

• [1] M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Romanian Rep. Phys., 67 (2015), 773--791


• [2] A. Atangana, A. Kilicman, On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative, Math. Probl. Eng., 2014 (2014), 9 pages


• [3] A. Atangana, S. C. Oukouomi Noutchie, Stability and Convergence of a Time-Fractional Variable Order Hantush Equation for a Deformable Aquifer, Abstr. Appl. Anal., 2013 (2013), 8 pages


• [4] A. Atangana, A. Secer, A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions, Abstr. Appl. Anal., 2013 (2013), 8 pages


• [5] C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692--703


• [6] G. R. J. Cooper, D. R. Cowan, Filtering using variable order vertical derivatives, Comput. Geosci., 30 (2004), 455--459


• [7] E. H. Doha, A. H. Bhrawy, D. Baleanu, S. S. Ezz-Eldien, The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation, Adv. Difference Equ., 2014 (2014), 14 pages


• [8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006)


• [9] R. Lin, F. Liu, V. Anh, I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), 435--445


• [10] C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57--98


• [11] A. Oustaloup, F. Levron, B. Matthieu, F. M. Nanot, Frequency-band complex noninteger differentiator: characterization and synthesis, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 47 (2000), 25--39


• [12] S. G. Samko, Fractional integration and differentiation of variable order, Anal. Math., 21 (1995), 213--236


• [13] T. H. Solomon, E. R. Weeks, H. L. Swinney, Observation of anomalous diffusion and Lévy ights in a two-dimensional rotating flow, Phys. Rev. Lett., 71 (1993), 3975--3978


• [14] H. Sun, W. Chen, Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physi. A, 388 (2009), 4586--4592


• [15] H. Sun, W. Chen, C. Li, Y. Chen, Finite difference schemes for variable-order time fractional diffusion equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 16 pages


• [16] D. Valério, J. S. Costa, Variable-order fractional derivatives and their numerical approximations, Signal Processing, 91 (2011), 470--483


• [17] G.-C. Wu, D. Baleanu, H.-P. Xie, S.-D. Zeng, Discrete Fractional Diffusion Equation of Chaotic Order, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 6 pages