Modified quasi‐boundary value method for the multidimensional nonhomogeneous backward time fractional diffusion equation

2019 ◽  
Author(s):  
Kokila Jayakumar
Keyword(s):  
Diffusion Equation ◽  
Boundary Value ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Boundary Value Method ◽  
Time Fractional Diffusion Equation

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.

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 On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
D. Goos ◽  
G. Reyero ◽  
S. Roscani ◽  
E. Santillan Marcus
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Real ◽  
Positive Semiaxis ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

We consider the time-fractional derivative in the Caputo sense of orderα∈(0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function inR+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit whenα↗1of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems whenα= 1, and the fractional diffusion equation becomes the heat equation.

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