Analytic Solution of Linear Fractional Differential Equation with Jumarie Derivative in Term of Mittag-Leffler Function

Uttam Ghosh^1, , Srijan Sengupta², Susmita Sarkar² and Shantanu Das^{3, 4}

¹Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India

²Department of Applied Mathematics, University of Calcutta, Kolkata, West Bengal, India

³Reactor Control System Design Section Bhabha Atomic Research Centre, Mumbai, India

⁴Department of Physics, Jadavpur University Kolkata, West Bengal, India

Cite this paper:
Uttam Ghosh, Srijan Sengupta, Susmita Sarkar and Shantanu Das. Analytic Solution of Linear Fractional Differential Equation with Jumarie Derivative in Term of Mittag-Leffler Function. American Journal of Mathematical Analysis. 2015; 3(2):32-38. doi: 10.12691/ajma-3-2-2

Abstract

There is no unified method to solve the fractional differential equation. The type of derivative here used in this paper is of Jumarie formulation, for the several differential equations studied. Here we develop an algorithm to solve the linear fractional differential equation composed via Jumarie fractional derivative in terms of Mittag-Leffler function; and show its conjugation with ordinary calculus. In these fractional differential equations the one parameter Mittag-Leffler function plays the role similar as exponential function used in ordinary differential equations.

Keywords:
jumarrie fractional derivative riemann-liouvelli fractional derivative mittag-leffler function fractional differential equations

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References:

[1]	de Oliveira, E. C., J. A. T. Machado. A review of definitions of fractional derivatives and Integral. Mathematical Problems in Engineering. Hindawi Publishing Corporation. 2014. 1-6.
[2]	Jumarie, G. Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions Further results, Computers and Mathematics with Applications, 2006. (51), 1367-1376.
[3]	G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” AppliedMathematics Letters, vol. 18, no. 7, pp. 817-826, 2005.
[4]	G. Jumarie, “An approach to differential geometry of fractional order via modified Riemann-Liouville derivative,” Acta Mathematica Sinica, vol. 28, no. 9, pp. 1741-1768, 2012.
[5]	G. Jumarie, “On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling,” Central European Journal of Physics, vol. 11, no. 6, pp. 617-633, 2013.
[6]	X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong Kong, 2011.
[7]	X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
[8]	Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations.John Wiley & Sons, New York, NY, USA; 1993.
[9]	Das. S. Functional Fractional Calculus 2nd Edition, Springer-Verlag 2011.
[10]	Podlubny I. Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. 1999; 198.
[11]	M. Caputo, “Linear models of dissipation whose q is almost frequency independent-ii,” Geophysical Journal of the Royal Astronomical Society, 1967. vol. 13, no. 5, pp. 529-539.
[12]	K M Kolwankar and A D. Gangal. Local fractional Fokker plank equation, Phys Rev Lett. 80 1998.
[13]	Abhay Parvate, A. D. Gangal. Calculus of fractals subset of real line: formulation-1; World Scientific, Fractals Vol. 17, 2009.
[14]	Abhay Parvate, Seema satin and A.D.Gangal. Calculus on a fractal curve in Rn arXiv:00906 oo76v1 3.6.2009; also in Pramana-J-Phys.
[15]	Abhay parvate, A. D. Gangal. Fractal differential equation and fractal time dynamic systems, Pramana-J-Phys, Vol 64, No. 3, 2005 pp 389-409.
[16]	E. Satin, Abhay Parvate, A. D. Gangal. Fokker-Plank Equation on Fractal Curves, Seema, Chaos Solitons & Fractals-52 2013, pp 30-35.
[17]	Erdelyi, A. Asymptotic expansions, Dover (1954).
[18]	Erdelyi, A. (Ed). Tables of Integral Transforms. vol. 1, McGraw-Hill, 1954.
[19]	Erdelyi, A. On some functional transformation Univ Potitec Torino 1950.
[20]	Mittag-Leffler, G. M.. Sur la nouvelle fonction Eα (x), C. R. Acad. Sci. Paris, (Ser. II) 137, 554-558 (1903).
[21]	Wiman, A. Uberden fundementalsatz in the theorie der funktionem Eα(z) Acta Math., 29. 1905. 191-201.
[22]	G. Jumarie. Fourier’s transformation of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivatives. J. Appl. Math. & informatics. 2008. 26. 1101-1121.