Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

By Mohammed Abdalla Al-Refai, Thabet Abdeljawad. Advances in Differential Equations, 2017

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In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/eb1ef345cdfe6cd8bd5f8fb95630f27d0edec760

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Citations in the corpus

Numerical investigations on COVID‐19 model through singular and non‐singular fractional operators (2020, background): “The arbitrary-order derivatives have very broad scope of utilizations such as in fields of engineering, physics, bio-mathematics [26, 27], signal processing, among others [1, 5, 19, 25, 30, 33, 41].”
New aspects of time fractional optimal control problems within operators with nonsingular kernel (2020, background): “Moreover, fractional calculus in continuous and discrete cases [33, 5, 4, 11, 7, 6, 1, 9, 2, 3] have been applied to physics [26], bioengineering [42], optics [24], fluid flow [21], energy systems [37] and biology [56, 54, 50, 47, 31, 30].”