Solution of Homogeneous Linear Fractional Differential Equations Involving Conformable Fractional Derivative

Authors

  • ANUJ TYAGI Research Scholar
  • Jyotsna Chandel

DOI:

https://doi.org/10.12723/mjs.63.1

Keywords:

Fractional exponential function, Riemann-Liouville derivative, Caputo fractional derivative, Conformable fractional derivative

Abstract

In this paper, we have found the solution of linear sequential fractional differential equations involving conformable fractional derivatives of order  with constant coefficients. For this purpose, we first discussed fundamental properties of the conformable derivative and then obtained successive conformable derivatives of the fractional exponential function. After this, we determined the analytic solution of linear sequential fractional differential equations (L.S.F.D.E.) in terms of a fractional exponential function. We have demonstrated this developed method with a few examples of homogeneous linear fractional differential equations. This method gives a conjugation with the method to solve classical linear differential equations with constant coefficients.

References

M. Caputo. “Linear models of dissipation whose q is alomancy independent”, Geophysical journal of the Royal Astronomical Society. 13. 5:529–539. (1967).

C. Baishya , “An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative”, SeMA Journal, 2022.

C. Baishya, “Dynamics of fractional Holling type-II predator-prey model with prey refuge and additional food to predator”, Journal of Applied Nonlinear Dynamics 10(2) (2021) 315--328.

C. Baishya, “A New Application of G’/G-Expansion Method for Travelling Wave Solutions of Fractional PDEs”, International Journal of Applied Engineering Research, 13(11) (2018) 9936-9942.

A. Kilbas. Saigo and R. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15 :31-4. (2004) http://dx.doi.org/10.1080/10652460310001600717

Li Z.B. and. He J.H. “Application of the fractional complex transform to fractional differential equations”, Nonlinear Science Letters A. 2:121-126. (2011)

R. Khalil, M. Al Horani , A. Yousef and M. Sababheh. “A new definition of fractional derivative, Journal of Computer”, Applied Mathematics, 264 (2014). https://doi.org/10.1016/j.cam.2014.01.002

B. Bayour, D. F. M. Torres. “Existence of solution to a local fractional nonlinear differential equation”, Journal of Computer. Applied Mathematics., (2016).

N. Benkhettou, S. Hassani and D. F. M. Torres. “A conformable fractional calculus on arbitrary time scales”, J. King Saud Univ. Sci., 28 ,93–98 (2016).

Lin S. and Lu. C. “Laplace transform for solving some families of fractional differential equations and its applications”, Advances in difference equations. 137. (2013).

http://dx.doi.org/10.1186/1687-1847-2013-137

Loonker, Deshna and P. K. Banerji. “Applications of Natural transform to differential Riemann-Liouville equations”, The Journal of the Indian Academy of Mathematics. 35.1: 151 -158. (2013).

U. Ghosh, S. Sengupta, S. Sarkar and S. Das. “Solution of System of Linear Fractional Differential Equations with Modified derivative of Jumarie Type”, Journal of Mathematical Analysis. 3.2: 32-38. (2015).

T. Abdeljawad, M. Al. Horani, R. Khalil. “Conformable fractional semigroups of operators”, J. Semigroup Theory Appl., 2015 (2015).

G. Jumarie. “An approach to differential geometry of fractional order via modified Riemann Liouville derivative”, Acta Mathematica Sinica. 28. 9: 1741–1768. (2012)

G. Jumarie. “Modified derivative and fractional Taylor series of non- differentiable functions Further results”, Computers and Mathematics with Applications. 51:1376. (2006)

G. Jumarie. “On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling”, Central European Journal of Physics. 11.6:617–633. (2013).

G. Jumarie. “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion”, Applied Mathematics Letters. 18. 7: 817–826. (2005)

G. Jumarie. “Table of some basic fractional calculus formulae derived from a modified Riemann Liouville derivative for non-differentiable functions”, Applied Mathematics Letters 22, 378-385. (2009).

Additional Files

Published

2023-01-14