scholarly journals Initial-boundary value problem for distributed order time-fractional diffusion equations

2019 ◽  
Vol 115 (1-2) ◽  
pp. 95-126 ◽  
Author(s):  
Zhiyuan Li ◽  
Yavar Kian ◽  
Éric Soccorsi
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Initial Boundary Value ◽  
Distributed Order

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 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.

scholarly journals The Solution of Initial Boundary Value Problem with Time and Space-Fractional Diffusion Equation via a Novel Inner Product

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Hulya Kodal Sevindir ◽  
Ali Demir
Keyword(s):  
Fourier Series ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Inner Product ◽  
Time And Space ◽  
Space Fractional Diffusion Equation ◽  
Initial Boundary Value

The main goal of this study is to find the solution of initial boundary value problem for the one-dimensional time and space-fractional diffusion equation which is a very intriguing topic for many researchers. With the aim of newly defined inner product, which is the main contribution of this study, the analytic solution of the boundary value problem is obtained. The time and space-fractional derivatives are defined in the Caputo sense which is more suitable than Riemann-Liouville sense. We apply the separation of variables method to reduce the problem to two separate fractional ODEs. The generalized solution is constructed/formed in the form of a Fourier series with respect to the eigenfunctions of a certain eigenvalue problem. In order to obtain the coefficients of the Fourier series for the solution, we define a new inner product which is the key point of study.

scholarly journals NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

2013 ◽  
Vol 54 (3) ◽  
pp. 153-170 ◽  
Author(s):  
RUNZHANG XU ◽  
YANBING YANG ◽  
SHAOHUA CHEN ◽  
JIA SU ◽  
JIHONG SHEN ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Wave Equations ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Wave Equations ◽  
Initial Boundary Value

AbstractThis paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

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.

Sign in / Sign up

Export Citation Format

Share Document