A generalized approach of fractional Fourier transform to stability of fractional differential equation
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Abstract
This research article deals with the Mittag-Leffler-Hyers-Ulam stability of linear and impulsive fractional order differential equation which involves the Caputo derivative. The application of the generalized fractional Fourier transform method and fixed point theorem, evaluates the existence, uniqueness and stability of solution that are acquired for the proposed non-linear problems on Lizorkin space. Finally, examples are introduced to validate the outcomes of main result.
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References
[1] A.A. Kilbas, Y. Luchko, H. Martinez and J.J. Trujillo, Fractional Fourier transform in the framework of fractional calculus operator, Integral Transforms and Special Functions, 21 (2010), 779–795. Google Scholar
[2] H. Dai, Z. Zheng and W. Wang, A new fractional wavelet transform, Communications in Non- linear Science Numerical Simulation, 44 (2017), 19–36. Google Scholar
[3] G.D. Medina, N.R. Ojeda, J.H. Pereira and L.G. Romero, Fractional Laplace transform and fractional calculus, International Mathematical Forum, 12 (2017), 991–1000. Google Scholar
[4] A. Kılıc ̧man and M. Omran, Note on fractional Mellin transform and applications, SpringerPlus, 5(1) (2016), 1–8. Google Scholar
[5] A. Prasad and V.K. Singh, The fractional Hankel transform of certain tempered distributions and pseudo-differential distributions, Annali Dell’universita’di Ferrara, 59(1) (2013), 141–158. Google Scholar
[6] N. Wiener, Hermition polynomial and Fourier analysis, Journal of Mathematics and Physics, 8 (1929), 70–73. Google Scholar
[7] H. Qusuay, H. Alqifiary and S.M. Jung, Laplace transform and generalized Hyers-Ulam stability of linear differential operators, Electronic Journal of Differential Equations, 80 (2014), 1–11. Google Scholar
[8] R. Murali, A. Ponmana Selvan and C. Park, Ulam stability of linear differential equation using Fourier transform method, AIMS Mathematics, 5 (2020), 766–780. Google Scholar
[9] B. Unyong, A. Mohanapriya, A. Ganesh, G. Rajchakit, V. Govindan, N. Gunasekaran and C.P. Lim, Fractional Fourier transform and stability of fractional differential equation on Lizorkin space, Advance in Difference Equation, 2020 (2020), no. 578. Google Scholar
[10] J. Wang and Y. Zhou, Mittag-Leffler-Ulam stability of fractional evolution equations, Applied Mathematics Letters, 25(4) (2012), 723–728. Google Scholar
[11] J. Wang and Y. Zhang, Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63(8) (2014), 1181–1190. Google Scholar
[12] J. Wang and X. Li, Eα-Ulam type stability of fractional order ordinary differential equations, Journal of Applied Mathematics and Computing, 45 (2014), 449–459. Google Scholar
[13] K. Liu, J. Wang, Y. Zhou and D.O. Regan, Hyers-Ulam stability and existence of solutions for fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons and Fractals, 132 (2020), 109–534. Google Scholar
[14] A. Zada, J. Alzabut, H. Waheed and L. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary condition, Advance in Difference Equation, 2020(2020), no. 64. Google Scholar
[15] S.O. Shah, A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic system with instantaneous and non instantaneous impulses on time scale, Applied Mathematics and Computation, 359 (2019), 202–213. Google Scholar
[16] J. Wang, A. Zada, W. Ali, Ulam’s type stability of first-order impulsive differential equations with variable in quasi-Banach spaces, International Journal of Nonlinear Science and Numerical Simulation, 19(5) (2018), 553–560. Google Scholar
[17] R. Agarwal, S. Hristova and D.O. Regan, Basic concepts of Riemann-Liouville fractional differential equation with non-instantaneous impulses, Symmetry, 11(5) (2019) 614. Google Scholar
[18] R. Almeida, A.B. Malinowska and M.T. Monteiro, Fractional differential equations with a Ca- puto derivative with respect to kernel function and their applications, Mathematical Methods in the Applied Sciences, 41 (2018), 336–352. Google Scholar
[19] A.K. Shukla and J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, Journal of Mathematical Analysis and Applications, 336 (2007), 797–811. Google Scholar
[20] K. Diethelm, An algorithm for the numerical solution of differential equation of fractional order, Electronic Transactions on Numerical Analysis, 5 (1997), 1–6. Google Scholar