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Zeitschrift für Analysis und ihre Anwendungen

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Volume 40, Issue 3, 2021, pp. 277–301
DOI: 10.4171/ZAA/1685

Published online: 2021-04-16

Maximum Principle for Space and Time-Space Fractional Partial Differential Equations

Mokhtar Kirane[1] and Berikbol T. Torebek[2]

(1) Khalifa University, Abu Dhabi, United Arab Emirates
(2) Inst. of Math. & Math. Modeling and Al-Farabi Kazakh National Univ., Almaty, Kazakhstan. and Ghent University, Belgium

In this paper, new estimates of the sequential Caputo fractional derivatives of a function at its extremum points are obtained.We derive comparison principles for the linear fractional differential equations, then apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion admits at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Y. Luchko [Fract. Calc. Appl. Anal. 14 (2011)]. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.

Keywords: Caputo derivative, sequential derivative, time-space fractional diffusion equation, fractional elliptic equation, maximum principle

Kirane Mokhtar, Torebek Berikbol: Maximum Principle for Space and Time-Space Fractional Partial Differential Equations. Z. Anal. Anwend. 40 (2021), 277-301. doi: 10.4171/ZAA/1685