New results on existence of Atangana-Baleanu fractional differential equations with dependence on the Lipschitz first derivatives

V. Pandiyammal; U. Karthik Raja

doi:10.26637/MJM0804/0084

Abstract

We study on the existence and uniqueness of solutions for a Atangana-Baleanu fractional differential equations with dependence on the Lipschitz first derivative conditions. We develop a Gronwall inequality in the frame of Atangana-Baleanu fractional integral. An example is given to illustrate the main results and investigate the stability in the sense of Ulam.is

Keywords:

Fractional differential equations, Atangana-Baleanu fractional derivative, Lipschitz first derivatives, Gronwall inequality, Ulam-Hyer stability

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

V. Pandiyammal Department of Mathematics, Arulmigu Palaniandavar College of Arts and Culture, Palani -624601, Tamil Nadu, India.
U. Karthik Raja Department of Mathematics, The Madura College, Madurai-625 011, Tamil Nadu, India.

Pages: 1834-1841
Date Published: 01-01-2020
Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

T. Abdeljawad, A Lyapunov type inequality for fractional operators with non- singular Mittag-Leffler kernel, J Inequalities Appl, 2017(130)(2017), 1-11.

T. Abdeljawad, QM. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J ComputAppl Math, 339(1)(2018), 218-230.

T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep Math Phys, 80(1)(2017), 11-27.

T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv Differ Equ, (2016), 1-18.

T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J Nonlinear Sci Appl, 10(3)(2017), 1098-1107.

Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad, On cauchy problems with Caputo Hadamard fractional derivatives, J Comput Anal Appl, 21(1)(2016), 661-681.

A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Therm Sci, 20(2)(2016), 763-769.

A. Atangana, JF. Gomez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, $mathrm{Nu}$ mer Methods Partial Differ Equ, 34(5)(2018), 1502-1523.

A. Atangana, JF. Gomez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102(2017), 285-294.

A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, (2016), 1-8.

Aziz Khan, Hasib Khan, J.F. Gomez-Aguilar, ThabetAbdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons and Fractals, 127(2019), 422-427.

Z. Ba, H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311(2005), 495-505.

${ }^{[13]}$ D. Baleanu, M. Inc, A. Yusuf, AI. Aliyu, Space-time fractional Rosenou-Haynam equation: lie symmetry analysis, explicit solutions and conservation laws, Adv Differ Equ, 2018(46)(2018), 1-14.

D. Baleanu, M. Inc, A. Yusuf, AI. Aliyu, Time fractional third-order evolution equation: symmetry analysis, explicit solutions and conservation laws, J Comput Nonlinear Dynam, 13(2)(2017), 021011.

U. Cakan, I. Ozdemir, An application of Krasnoselskii fixed point theorem to some nonlinear functional integral equations, NevsehirBilimveTeknolojiDergisi $3(2)(2014), 66-73$.

JD. Djida, A. Atangana, I. Area, Numerical computation of fractional derivative with non-local and non-singular kernel, Math Model Nat Phenom, 12(3)(2017), 4-13.

YY. Gambo, F. Jarad, T. Abdeljawad, D. Baleanu, On Caputo modification of the Hadamard fractional derivativ, Adv Differ Equ., 2014(10)(2014), 1-12.

L. Gaul, P. Klein, S. Kempe, Damping description involving fractional operators, Mech.Systems Signal Processing, 5(1991), 81-88.

M. Donatelli, M. Mazza, S.S. Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phy., 307(2016), 262279.

F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of AtaganaBaleanu derivative, Chaos Solitons Fractals, 117(2018), $16-20$

F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv Differ Equ, 2012(142)(2012), 1-8.

Daniela Lera, Yaroslav D Sergeyev, Acceleration of univariate global optimization algorithms working with lipschitz functions and lipschitz first derivatives, SIAM J. Optim, 23(1), 508-529

H. Li, Y. Jiang, Z. Wang, L. Zhang, Z. Teng, Global Mittag-Leffler stability of coupled system of fractionalorder differential equations on network, Appl. Math. Comput., 270(2015), 269-277.

I. Koca, Analysis of rubella disease model with nonlocal and non-singular fractional derivative, Int J Optim Control, 8(1)(2018), 17-25.

A. Kilbas, HM. Srivastava, JJ. Trujillo, Theory and application of fractional differential equations, North Holland Mathematics Studies, 204(2006).

I. Koca, A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Frbrizio and Atagana-Baleanu fractional derivatives, Therm Sci, 21(6A)(2017), 2299-2305.

I. Koca, Modelling the spread of ebola virus with Atangana-Baleanu fractional operators, Eur Phys J Plus 133(3)(2018), 1-11.

E. Dmitri Kvasov, D. Yaroslav Sergeyev, A univariate global search working with a set of Lipschitz constants for the first derivative, Optim Lett, 3(2009), 303-318

T. Mekkaoui, A. Atangana, New numerical approximation of fractional derivative with non-local and nonsingular kernel: application to chaotic models, Eur Phys J Plus, 132(10)(2017), 4.

KM. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133(15)(2018), 1-13.

NdolaneSene, Stokes first problem for heated flat plate with Atangana-Baleanu fractional derivative, Chaos, Solitons and Fractals, 117(2018), 68-75.

SG. Samko, AA. Kilbas, IO. Marichev, Fractional inte-grals and derivatives: theory and applications. In: Gordon and Breach, Yverdon; (1993).

A. Yusuf, M. Inc, AI. Aliyu, D. Baleanu, Efficiency of the new fractional derivative with nonsingularMittag Leffler kernel to some nonlinear partial differential equations, Chaos Solitions Fractals, 116(2018), 220-226.