On the generalization of right fractional order derivatives and the Darboux problem for partial differential equations

Djalal Boucenna, Faicel Missaoui, Abdellatif Ben Makhlouf

Abstract


In this work, we explore the existence and uniqueness of solutions to the Darboux problem for partial differential equations (DPPDEs) involving the right partial Caputo $\vartheta$-fractional derivative. The Ulam-Hyers stability (UHS) of DPPDE is also studied by using a generalized Gronwall inequality. Finally, we illustrate our results by an example.

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References


M. Nadeem, L.F. Iambor, Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics, Fractal Fract. 8 (2024), 277.

P. Kumar, S.M. Sivalingam, V. Govindaraj, Forecasting of HIV/AIDS in South Africa using 1990 to 2021 data: novel integer-and fractional-order fittings, International Journal of Dynamics and Control 12 (2024), 2247-2263.

A.M. Nagy, S. Assidi, A. Ben Makhlouf, Convergence of solutions for perturbed and unperturbed cobweb models with generalized Caputo derivative, Boundary Value Problems 2022 (2022), 89.

Z. Korichi, A. Souigat, R. Bekhouche, M.T. Meftah, Solution of the fractional Liouville equation by using Riemann-Liouville and Caputo derivatives in statistical mechanics, Theoretical and Mathematical Physics 218 (2024), 336-345.

P. Kalamani, A. Raj, P. Kumar, Results on the existence, uniqueness, and controllability of neutral fractional differential equations in the sense of Atangana-Baleanu derivative, The Journal of Analysis 2023 (2023), 685.

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York, 2015.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, 1999.

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation 44 (2017), 460-481.

S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with _nite delay, Nonlinear Analysis: Hybrid Systems 3 (2009), 597-604.

S. Abbas, M. Benchohra, Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations, Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 51 (2012), 5-18.

S. Abbas, M. Benchohra, On The Generalized Ulam-Hyers-Rassias Stability For Darboux Problem For Partial Fractional Implicit Differential Equations, Applied Mathematics E-Notes 14 (2014), 20-28.

S. Abbas, M. Benchohra, Ulam-Hyers Stability for the Darboux Problem for Partial Fractional Differential and Integro-differential Equations via Picard Operators, Results in Mathematics 65 (2014), 67-79.

H. Arfaoui, A. Ben Makhlouf, Some results for a class of delayed fractional partial differential equations with Caputo-Hadamard derivative, Mathematical Methods in the Applied Sciences 46 (2023), 9954-9965.

H. Fan, B. Zheng, Some new generalized Gronwall-Bellman type inequalities arising in the theory of fractional di_erential-integro equations, WSEAS Transactions on Mathematics 13 (2014), 820-829.

H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics 2011 (2011), Article ID 298628, 51 pages.




DOI: https://doi.org/10.52846/ami.v53i1.2065