Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

By Mohammed Abdalla Al-Refai, Yuri Luchko. 2014

63 citations recorded by the source corpus.

In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/bbfe03cc394f248d93bb3cd38bbf507f7ec8168e

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications (2016, background): “For left-side Riemann-Liouville fractional derivatives, Al-Refai and Luchko in [1, 2] obtained a similar result as L 0D η t0f(t0) ≥ t−η 0 Γ(1− η)0 Theorem 3.”
A survey of useful inequalities in fractional calculus (2017, unknown): “Similar results also appeared in the papers of Al-Refai and Luchko [2], [3].”
On some new properties of fractional derivatives with Mittag-Leffler kernel (2017, background): “For example, they can be used to model controllability [22], viscoelastic flows [25, 12], chaotic systems [33], Stokes problems [46], thermoelasticity [35], several vibration and diffusion processes [8, 39, 6], bioengineering problems [24], and many other complex phenomena.”