Related papers: Initial boundary value problems for time-fractiona…
We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with…
In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 0<x<L, \ t>0 \qquad (1) \end{equation} subject…
We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for…
In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value…
In this paper, we study the initial-boundary value problem of the Navier-Stokes system in the half space. We prove the unique solvability of the weak solution on some short time interval (0, T) with the velocity in $C^{\alpha, \frac12…
We consider a linear equation $\partial_t u = \mathcal{L}u$, where $\mathcal{L}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field…
The initial-boundary value problem for the Kundu--Eckhaus equation on the half-line is considered in this paper by using the Fokas method. We will show that the solution $u(x,t)$ can be expressed in terms of the solution of a matrix…
A rigorous methodology for the analysis of initial boundary value problems on the half-line, $0<x<\infty$, $t>0$, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the…
In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],…
This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized…
In the article, in a rectangular domain, by the Fourier method, the initial boundary value problem for a high-order equation with two lines of degeneracy with a fractional derivative in the sense of Caputo is investigated for uniqueness and…
A boundary value problem for a fractional power $0 < \varepsilon < 1$ of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when $\varepsilon \rightarrow 0$. It is solved numerically using a…
The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann-Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space.…
In this note we propose the exact artificial boundary conditions formula to the fractional reaction-subdiffusion equation on an unbounded domain. With the application of Laplace transformation, the exact artificial boundary conditions…
In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the…
In this paper, we discuss the uniqueness for solution to time-fractional diffusion equation $\partial_t^\alpha (u-u_0) + Au=0$ with the homogeneous Dirichlet boundary condition, where an elliptic operator $-A$ is not necessarily symmetric.…
We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of…
We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a…
In this paper, we consider initial-boundary value problems for two-component nonlinear systems of time-fractional diffusion equations with the homogeneous Neumann boundary condition and non-negative initial values. The main results are the…