Maximum principle for space and time-space fractional partial differential equations

Kirane M. Torebek B.T.
2021 European Mathematical Society Publishing House

Zeitschrift für Analysis und ihre Anwendungen
2021 #40 Issue 3 277 - 301 pp.

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.

Caputo derivative , Fractional elliptic equation , Maximum principle , Sequential derivative , Time-space fractional diffusion equation

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Department of Mathematics, College of Arts and Sciences, Khalifa University, P.O. Box 127788, Abu Dhabi, United Arab Emirates
NAAM Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., Almaty, 050010, Kazakhstan
Al-Farabi Kazakh National University, Al-Farabi Ave. 71, Almaty, 050040, Kazakhstan
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Ghent, Belgium

Department of Mathematics
NAAM Research Group
Institute of Mathematics and Mathematical Modeling
Al-Farabi Kazakh National University
Department of Mathematics: Analysis

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