scholarly journals Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

2018 ◽  
Vol 81 ◽  
pp. 14-20 ◽  
Author(s):  
Meiirkhan Borikhanov ◽  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Maximum Principle ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Dirichlet Conditions ◽  
Fractional Diffusion Equations

scholarly journals The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

2020 ◽  
Vol 8 ◽  
Author(s):  
Guangming Xue ◽  
Funing Lin ◽  
Guangwang Su
Keyword(s):  
Maximum Principle ◽  
Fractional Derivatives ◽  
Arbitrary Point ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

scholarly journals Maximum principle and its application for the time-fractional diffusion equations

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Yury Luchko
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
A Priori Estimates ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations ◽  
Generalized Time

AbstractIn 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.

scholarly journals Strong maximum principle for fractional diffusion equations and an application to an inverse source problem

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Yikan Liu ◽  
William Rundell ◽  
Masahiro Yamamoto
Keyword(s):  
Maximum Principle ◽  
Parabolic Equations ◽  
Fractional Diffusion ◽  
Uniqueness Result ◽  
Strong Maximum Principle ◽  
Diffusion Equations ◽  
Inverse Source Problem ◽  
Fractional Diffusion Equations ◽  
Parabolic Case

AbstractThe strong maximum principle is a remarkable property of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak maximum principle, in this paper we establish a strong maximum principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.

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