scholarly journals Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations

2021 ◽  
Vol 24 (1) ◽  
pp. 168-201
Author(s):  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Well Posedness ◽  
Initial Boundary Value Problems ◽  
Fractional Diffusion Equations ◽  
Initial Boundary Value ◽  
Weak And Strong Solutions

Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Mohammed Al-Refai ◽  
Yuri Luchko
Keyword(s):  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Initial Boundary Value Problems ◽  
Fractional Diffusion Equations ◽  
Non Linear ◽  
Initial Boundary Value

AbstractIn this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Zhiyuan Li ◽  
Yuri Luchko ◽  
Masahiro Yamamoto
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Initial Boundary Value Problems ◽  
Fractional Diffusion Equations ◽  
Initial Boundary Value ◽  
Distributed Order

AbstractThis article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1. Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.

scholarly journals Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
The Maximum Principle ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

scholarly journals Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

2018 ◽  
Vol 21 (2) ◽  
pp. 276-311 ◽  
Author(s):  
Adam Kubica ◽  
Masahiro Yamamoto
Keyword(s):  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Unique Existence ◽  
Fractional Diffusion Equations ◽  
Time Variables ◽  
Initial Boundary Value

Abstract We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

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