On generalized weighted fractional order derivatives and Darboux problem for partial differential equations

Wiam Bentimama, Djalal Boucenna, Hassane Khellaf, Mohamed Rhaima, Lassaad Mchiri, Abdellatif Ben Makhlouf

Abstract


This paper unveils a novel mathematical construct, namely the weighted fractional derivative, and delves into its comprehensive exploration. By formulating pertinent hypotheses concerning the source term, the manuscript not only verifies the existence, uniqueness, and stability of solutions for Darboux problems but also introduces the transformative aspect of the weighted fractional operator in this context.

Full Text:

PDF

References


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

S. Abbas, M. Benchohra, Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Analysis: Hybrid Systems 4 (2010), 406-413.

S. Deb, A. Das, Application of a generalization of Darbos fixed point theorem on an integral equation involving the weighted fractional integral with respect to another function, Mathematics, arXiv:2306.08922, 2023.

H. Arfaoui, A. Ben Makhlouf, Some results for a class of two-dimensional fractional hyperbolic differential systems with time delay, Journal of Applied Mathematics and Computing 68 (2022), 2389-2405.

M. Benchohra, J.E. Lazreg, Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions, Rom. J. Math. Comput. Sci 4 (2014), no. 1, 60-72.

A. Ben Makhlouf, D. Boucenna, Ulam-Hyers-Rassias Mittag-Leffler stability for the Darboux problem for partial fractional differential equations, Rocky Mountain Journal of Mathematics 51 (2021), 1541-1551.

M. Bohner, O. Tunç, C. Tunç, Qualitative analysis of Caputo fractional integro-differential equations with constant delays, Comput. Appl. Math. 214 (2021), 3457-3471.

D. Boucenna, D. Baleanu, A. Ben Makhlouf, A.M. Nagy, Analysis and numerical solution of the generalized proportional fractional Cauchy problem, Applied Numerical Mathematics 167 (2021), 173-186.

A. Boutiara, A coupled system of nonlinear Langevin Fractional q-Difference equations associated with two different fractional orders in Banach space, Kragujevac Journal of Mathematics 48 (2024), 555-575.

M. Samraiz, M. Umer, A. Kashuri, T. Abdeljawad, S. Iqbal, N. Mlaiki, On Weighted (k; s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation, Fractal & Fractional 5 (2021), 118.

H.V.S. Chauhan, B. Singh, C. Tunç, O. Tunç, On the existence of solutions of non-linear 2D Volterra integral equations in a Banach Space, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), 101.

C. Derbazi, Solvability for a class of nonlinear Caputo-Hadamard fractional differential equations with p-Laplacian operator in Banach spaces, Facta Universitatis. Series: Mathematics and Informatics 2020 (2020), 693-711.

C. Derbazi, Nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with Dirichlet Boundary conditions in Banach spaces, Kragujevac Journal of Mathematics 46 (2022), 841-855.

H. Fan, B. Zheng, Some new generalized Gronwall-Bellman type inequalities arising in the theory of fractional differential-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, J. of App. Math. 2011 (2011), Article ID 98628, 51 pages.

L. He, Y. Wang, Y. Wei, M. Wang, X. Hu, Q. Shi, An adaptive central difference Kalman filter approach for state of charge estimation by fractional order model of lithium-ion battery, Energy 244 (2022), no. 3, Article ID 122627.

F. Jarad, T. Abdeljawad, K. Shah, On the Weighted Fractional Operators of a Function with Respect to Another Function, Fractals 28 (2020), Article ID 2040011.

P.D. Lax, R.D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (1956), no. 2, 267-293.

Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2007), 1533-1552.

B. Ramakrishnan, M.E. Cimen, A. Akgul, C. Li, K. Rajagopal, H. Kor, Chaotic Oscillations in a Fractional-Order Circuit with a Josephson Junction Resonator and Its Synchronization Using Fuzzy Sliding Mode Control, Mathematical Problems in Engineering 2022 (2022), Article ID 6744349.

O. Tunç, Ö. Atan, C. Tunç, J.C. Yao, Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov-Razumikhin Method, Axioms 10 (2021), Article ID 58.

A.N. Vityuk, A.V. Golushkov, Darboux problem for a differential equation with fractional derivative, Nonlinear Oscillations 8 (2005), 450-462.

A.N. Vityuk, A.V. Mikhailenko, On one class of differential equations of fractional order, Non-linear Oscillations 11 (2008), 307-319.

L. Viviani, M. Di Paola, G. Royer-Carfagni, A fractional viscoelastic model for laminated glass sandwich plates under blast actions, International Journal of Mechanical Sciences 222 (2022), Article ID 107204.

A. Zeb, P. Kumar, V.S. Erturk, T. Sitthiwirattham, A new study on two different vaccinated fractional-order COVID-19 models via numerical algorithms, Journal of King Saud University - Science 34 (2022), Article ID 101914.




DOI: https://doi.org/10.52846/ami.v52i2.1970