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.

General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Yuri Luchko ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value ◽  
General Time

AbstractIn this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

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.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

A class of densely invertible parabolic operator equations

1969 ◽  
Vol 1 (3) ◽  
pp. 363-374 ◽  
Author(s):  
R.S. Anderssen
Keyword(s):  
Variational Methods ◽  
Variational Formulation ◽  
Lagrange Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Parabolic Differential Equations ◽  
Operator Equations ◽  
Initial Boundary Value

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

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