Existence and stability for fractional differential equations
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Date
2025
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Université Badji Mokhtar Annaba
Abstract
This thesis investigates the theoretical properties and applications of the Riesz-Caputo fractional derivative. Key contributions include the rigorous analysis of various boundary value problems (BVPs), such as multi-point BVPs with integral boundary conditions, pantograph-type delay differential equations, and a fractional thermostat model with nonlocal boundary conditions. For these problems, criteria for the existence, uniqueness, positivity, and Ulam-Hyers stability of solutions are established using advanced fixed-point theorems (Krasnoselskii, Leray-Schauder Alternative, Schaefer, Guo-Krasnoselskii) and detailed Green's function analysis. The work highlights the suitability of the symmetric Riesz- Caputo operator for modeling systems with bidirectional memory effects and non-local interactions, where its Caputo-type formulation facilitates the use of physically interpretable initial/boundary conditions. The thesis also explores the conceptual underpinnings of fractional calculus, including its historical development and potential links to fractal geometry, and surveys applications in fields like viscoelasticity and anomalous diffusion. Numerical examples complement the theoretical findings. The research aims to advance the understanding of this specific fractional operator and its role in mathematical modeling of complex phenomena.
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Keywords
fractional calculus; riesz-caputo derivative; boundary value problems; fixed point theorems; stability analysis; memory effects; pantograph equations; thermostat model