# Maximum Principle and Its Application for the Time-Fractional Diffusion Equations

• Volume: 14, Issue: 1, page 110-124
• ISSN: 1311-0454

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## Abstract

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MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.

## How to cite

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Luchko, Yury. "Maximum Principle and Its Application for the Time-Fractional Diffusion Equations." Fractional Calculus and Applied Analysis 14.1 (2011): 110-124. <http://eudml.org/doc/219664>.

@article{Luchko2011,
abstract = {MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.},
author = {Luchko, Yury},
journal = {Fractional Calculus and Applied Analysis},
keywords = {Time-Fractional Diffusion Equation; Time-Fractional Multiterm Diffusion Equation; Time-Fractional Diffusion Equation of Distributed Order; Extremum Principle; Caputo Fractional Derivative; Generalized Riemann-Liouville Fractional Derivative; Initial-Boundary-Value Problems; Maximum Principle; Uniqueness Results; time-fractional diffusion equation; time-fractional multi-term diffusion equation; time-fractional diffusion equation of distributed order; extremum principle; Caputo fractional derivative generalized Riemann-Liouville fractional derivative initial-boundary-value problems maximum principle uniqueness results},
language = {eng},
number = {1},
pages = {110-124},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Maximum Principle and Its Application for the Time-Fractional Diffusion Equations},
url = {http://eudml.org/doc/219664},
volume = {14},
year = {2011},
}

TY - JOUR
AU - Luchko, Yury
TI - Maximum Principle and Its Application for the Time-Fractional Diffusion Equations
JO - Fractional Calculus and Applied Analysis
PY - 2011
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 14
IS - 1
SP - 110
EP - 124
AB - MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.
LA - eng
KW - Time-Fractional Diffusion Equation; Time-Fractional Multiterm Diffusion Equation; Time-Fractional Diffusion Equation of Distributed Order; Extremum Principle; Caputo Fractional Derivative; Generalized Riemann-Liouville Fractional Derivative; Initial-Boundary-Value Problems; Maximum Principle; Uniqueness Results; time-fractional diffusion equation; time-fractional multi-term diffusion equation; time-fractional diffusion equation of distributed order; extremum principle; Caputo fractional derivative generalized Riemann-Liouville fractional derivative initial-boundary-value problems maximum principle uniqueness results
UR - http://eudml.org/doc/219664
ER -

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