Current Developments in Mathematical Sciences

Current Developments in Mathematical Sciences

Volume: 1

Frontiers in Fractional Calculus

This book brings together eleven topics on different aspects of fractional calculus in a single volume. It provides readers the basic knowledge of fractional calculus and introduces advanced topics ...
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A Survey on the Recent Results Regarding Maximum Principles for the Time-Fractional Diffusion Equations

Pp. 33-69 (37)

Yuri Luchko and Masahiro Yamamoto

Abstract

In this chapter, a survey on the recent results regarding the maximum principles for the time-fractional diffusion equations is presented. We formulate the maximum principles for the time-fractional partial differential equations with both the Caputo fractional derivative and the Riemann-Liouville fractional derivative. Along with the single-term time-fractional differential equations, the multi-term equations and the equations of the distributed order are considered. We also discuss some important applications of the maximum principles for the time-fractional diffusion equations including a priori estimates for solutions of the initial-boundaryvalue problems for these equations and uniqueness of their solutions.

Keywords:

Fractional derivatives, Time-fractional diffusion equation, Initialboundary- value problems, Maximum principle, A priori estimates, Uniqueness of solutions.

Affiliation:

Department of Mathematics, TU of Applied Sciences Berlin, Germany.