Existence and uniqueness of the weak solution of initial-boundary-value problems for the time-space fractional diffusion equation

2017 ◽  
Author(s):  
Yanhua Wen ◽  
Xuanxuan Xi ◽  
Jun Wang
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Time Space ◽  
Space Fractional Diffusion Equation ◽  
Initial Boundary Value

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 a fractional differential equation with hyper-Bessel operator

2018 ◽  
Vol 21 (1) ◽  
pp. 200-219 ◽  
Author(s):  
Fatma Al-Musalhi ◽  
Nasser Al-Salti ◽  
Erkinjon Karimov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Bessel Differential Operator ◽  
Fractional Differential ◽  
Inverse Source Problems ◽  
Initial Boundary Value

AbstractDirect and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.

scholarly journals TWO INITIAL-BOUNDARY VALUE PROBLEMS WITH NONLINEAL BOUNDARY CONDITIONS FOR ONE-DIMENSION HYPERBOLIC EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 46-56
Author(s):  
L.S. Pulkina ◽  
M.V. Strigun
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Boundary Conditions ◽  
One Dimension ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

In this paper, the initial-boundary value problems for hyperbolic equationwith nonlinear boundary conditions are considered. Existence and uniqueness ofgeneralized solution are proved.

Sign in / Sign up

Export Citation Format

Share Document