Mohammed Said SOUID, Zoubida BOUAZZA, Ali YAKAR

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

This paper investigates the sufficient conditions for the existence and uniqueness of a class of Riemann-Liouville fractional differential equations of variable order with fractional boundary conditions. The problem is converted into differential equations of constant orders by combining the concepts of generalized intervals and piecewise constant functions. We derive the required conditions for ensuring the uniqueness of the problem in order to utilize the Banach fixed point theorem. The stability of the obtained solution in the Ulam-Hyers-Rassias (UHR) sense is also investigated, and we finally provide an illustrative example.

Keywords:

Boundary value problem, Stability Existence, Uniqueness, Fractional calculus,

PDF

___

A. A. Kilbas, H. M Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
S. G. Samko, B. Boss, Integration and Differentiation to a Variable Fractional Order, Integral Transforms and Special Functions 1 (1993) 277–300.
S. G. Samko, Fractional Integration and Differentiation of Variable Order, Analysis Mathematica 21 (1995) 213–236.
C. F. Lorenzo, T. T. Hartley, Variable Order and Distributed Order Fractional Operators, Nonlinear Dynamics 29 (2002) 57–98.
C. F. Lorenzo, T. T. Hartley, Initialization, Conceptualization, and Application in the Generalized (Fractional) Calculus, Critical Reviews in Biomedical Engineering 35(6) (2007) 447–553.
A. Abirami, P. Prakash, Y-K. Ma, Variable-Order Fractional Diffusion Model-Based Medical Image Denoising, Mathematical Problems in Engineering (2021) Article ID 8050017 10 pages.
J. F. Gomez-Aguilar, Analytical and Numerical Solutions of Nonlinear Alcoholism Model via Variable-Order Fractional Differential Equations, Physica A: Statistical Mechanics and its Applications 494 (2018) 52–57.
C. F. M. Coimbra, Mechanics with Variable-Order Differential Operators, Annalen der Physik 12 (11-12) (2003) 692–703.
M. Di Paola, G. Alotta, A. Burlon, G. Failla, A Novel Approach to Nonlinear Variable-Order Fractional Viscoelasticity, Philosophical Transactions of the Royal Society A 378 (2020) 20190296.
M. H. Heydari, Z. Avazzadeh, A New Wavelet Method for Variable-Order Fractional Optimal Control Problems, Asian Journal of Control 20(5) (2018) 1804–1817.
A. D. Obembe, M. D. Hossain, S. A. Abu-Khamsin, Variable-Order Derivative Time Fractional Diffusion Model for Heterogeneous Porous Media, Journal of Petroleum Science and Engineering 152 (2017) 391–405.
H. Sun, A. Chang, Y. Zhang, W. Chen, A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications, Fractional Calculus and Applied Analysis 22 (2019) 27–59.
H. Sun, W. Chen, Y. Chen, Variable-Order Fractional Differential Operators in Anomalous Diffusion Modeling, Physica A: Statistical Mechanics and Its Applications 388(21) (2009) 4586–4592.
A. Akgül, M. Inc, D. Baleanu, On Solutions of Variable-Order Fractional Differential Equations, An International Journal of Optimization and Control: Theories and Applications (IJOCTA) 7(1) (2017) 112–116.
R. Lin, F. Liu, V. Anh, I. Turner, Stability and Convergence of a New Explicit Finite-Difference Approximation for the Variable-Order Nonlinear Fractional Diffusion Equation, Applied Mathematics and Computation 212 (2) (2009) 435–445.
D. Tavares, R. Almeida, D. F. M. Torres, Caputo Derivatives of Fractional Variable Order Numerical Approximations, Communications in Nonlinear Science and Numerical Simulation 35(2016) 69–87.
D. Valerio, J. S. Costa, Variable-Order Fractional Derivatives and Their Numerical Approximations, Signal Process 91 (3) (2011) 470–483.
S. Zhang, S. Sun, L. Hu, Approximate Solutions to Initial Value Problem for Differential Equation of Variable Order, Journal of Fractional Calculus and Applications 9 (2) (2018) 93–112.
A. Refice, M. S. Souid, A. Yakar, Some Qualitative Properties of Nonlinear Fractional Integro-Differential Equations of Variable Order, An International Journal of Optimization and Control: Theories and Applications (IJOCTA) 11(3) (2021) 68–78.
A. Benkerrouche, D. Baleanu, M. S. Souid, A. Hakem, M. Inc, Boundary Value Problem for Nonlinear Fractional Differential Equations of Variable Order via Kuratowski Mnc Technique, Advances in Difference Equations 365 (2021) 19 pages.
A. Benkerrouche, M. S. Souid, S. Chandok, A. Hakem, Existence and Stability of a Caputo Variable-Order Boundary Value Problem, Journal of Mathematics Article ID 7967880 (2021) 16 pages.
A. Benkerrouche, M. S. Souid, K. Sitthithakerngkiet, A. Hakem, Implicit Nonlinear Fractional Differential Equations of Variable Order, Boundary Value Problems 2021 (64) (2021) 16 pages.
Z. Bouazza, S. Etemad, M. S. Souid, S. Rezapour, F. Martinez, M. K. A. Kaabar, A Study on the Solutions of a Multiterm Fractional Boundary Value Problem of Variable Order, Journal of Function Spaces Article ID 9939147 (2021) 9 pages.
Z. Bouazza, M. S. Souid, H. Günerhan, Multiterm Boundary Value Problem of Caputo Fractional Differential Equations of Variable Order, Advances in Difference Equations 400 (2021) 17 pages.
A. Benkerrouche, M. S. Souid, F. Jarad, A. Hakem, On Boundary Value Problems of Caputo Fractional Differential Equation of Variable Order via Kuratowski Mnc Technique, Advances in Continuous and Discrete Models 43 (2022) 19 pages.
A. Razminia, A. F. Dizaji, V. J. Majd, Solution Existence for Nonautonomous Variable-Order Fractional Differential Equations, Mathematical and Computer Modelling 55 (3-4) (2012) 1106–1117.
A. Yakar, M. E. Koksal, Existence Results for Solutions of Nonlinear Fractional Differential Equations, Abstract and Applied Analysis Article ID 267108 (2021) 12 pages.
S. Zhang, Existence of Solutions for Two Point Boundary Value Problems with Singular Differential Equations of Variable Order, Electronic Journal of Differential Equations 245 (2013) 1–16.
Z. Akdoğan, A. Yakar, M. Demirci, Discontinuous Fractional Sturm–Liouville Problems with Transmission Conditions, Applied Mathematics and Computation 350 (2019) 1–10.
A. Yakar, H. Kutlay, A Note on Comparison Results for Fractional Differential Equations, AIP Conference Proceedings 1676 (2015) Article ID 020064 5 pages.
H. Afshari, M. S. Abdo, J. Alzabut, Further Results on Existence of Positive Solutions of Generalized Fractional Boundary Value Problems, Advances in Difference Equations 600 (2020) 13 pages.
A. Seemab, M. Ur Rehman, J. Alzabut, A. Hamdi, On the Existence of Positive Solutions for Generalized Fractional Boundary Value Problems, Boundary Value Problems 186 (2019) 20 pages.
I. A. Rus, Ulam Stabilities of Ordinary Differential Equations in a Banach Space, Carpathian Journal of Mathematics 26 (1) (2010) 103–107.
D. R. Smart, Fixed Point Theorems, Cambridge University Press, 66, 1980.