Abel Fractional Differential Equations Using Variation Of Parameters Method

Authors

  • Nithya Devi Department of Mathematics, Periyar University, Salem-636011, INDIA
  • P. Prakash Department of Mathematics, Periyar University, Salem-636011, INDIA

DOI:

https://doi.org/10.23755/rm.v42i0.723

Abstract

The Variation of Parameters Method (VPM) is utilized throughout the research to identify a numerical model for a nonlinear fractional Abel differential equation (FADE). The approach given here is used to solve the initial problem of fractional Abel differential equations. There is no conversion, quantization, disturbance, structural change, or precautionary concerns in the proposed method, although it is easy with numerical solutions. The measured values are graphed and tabulated to be compared with the numerical model.

References

A. Khastan, J. J. Nieto and R. Rodriguez-Lopez,” Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty”, Fixed Point Theory and Applications, 21(2014).

A.Kilbas, H. Srivastava and J. Trujillo,” Theory and Applications of Fractional Differential Equations”, vol.204 of North-Holland Mathematical studies, Elsevier, New York, 2006.

I. Podlubny ” Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Pree, New York, 1999.

S. G. Samko, A. A. Kilbas and O. I. Marichev, ” Fractional Integrals and Derivatives”, Theory and Applications, Gordon and Breach Science Publishers, India, 1993.

S. Salahshour, T. Allahviranloo et.al, ” Solving fuzzy fractional differential equations by fuzzy Laplace transform”, Communications in Nonlinear Science and Numerical Simulation, 17(2012) 1372-1381.

G. M. Zaslavsky, ”Hamiltonian chaos and fractional dynamics ”, Oxford University Press on Demand, 2005.

M. Merdan, A. Yildirim, and A. Gokdogan, ”Numerical solution of time-fraction modified equal width wave equation”, Engineering Computations, 29(7), 2012.

S. T. Mohyud-Din, M.A. Noor and A.Waheed, ”Variation of Parameters method for initial and boundary value problems 1”, Journal of World Applied Sciences, 11(2010),622-639. 15

G. Jumarie, ”Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative”, Applied Mathematics Letters, 22(11),(2009)1659-1664.

M. A. Noor, S. T. Mohyud-Din, and A.Waheed, ”Variation of Parameters method for solving fifth-order boundary value problems”, Applied Mathematics and Information Sciences, 2(2) (2008) 135-141.

M. Hao and C. Zhai, Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order, J. Nonlinear Sci. Appl. 7 (2014), 131–137.

R. Khalil, M. Al Horani, A. Yousef. and M. Sababheh, A new Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014), pp. 65–70.

K. S. Miller, An introduction to fractional calculus and fractional differential equations, J.Wiley, and Sons, New York, (1993).

J. A. Nanware and D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, J. Nonlinear Sci. Appl. 7 (2014), 246–254. 16

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Published

2022-06-30

Issue

Vol. 42 (2022)

Section

Volume 42-2022

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