Related papers: Inverse problem for the subdiffusion equation with…
Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order $ \al \in (0,1) $ is taken in the Caputo's sense. The first one has a constant condition on $ x = 0 $ and the second presents a…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…
Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The…
The paper addresses the formulation and analysis of direct and inverse problems for a Langevin-type fractional differential equation under a non-local condition imposed on the time variable. An additional condition for solving the inverse…
In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive…
Let $u_t-a(t)u_{xx}=f(x, t)$ in $0\leq x \leq \pi,\,\,t\geq 0.$ Assume that $u(0,t)=u_1(t)$, $u(\pi,t)=u_2(t)$, $u(x,0)=h(x)$, and the extra data $u_x(0,t)=g(t)$ are known. The inverse problem is: {\it How does one determine the unknown…
We present a numerical procedure of solving the subdiffusion equation with Caputo fractional time derivative. On the basis of few examples we show that the subdiffusion is a 'long time memory' process and the short memory principle should…
We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$--subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the…
Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional…
Inverse problem for multi-term fractional parabolic equation in two dimensional space, involving m + 1 Caputo fractional derivatives in time, is investigated. Presence of nonlocal boundary conditions leads to a non-self-adjoint spectral…
The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation…
The Schr\"odinger equation $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) + f(x,t)$ ( $0<t\leq T, \, 0<\rho<1$), with the Riemann-Liouville derivative is considered. An inverse problem is investigated in which, along with $u(x,t)$, also a…
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
We present a set of restrictions on the fractional differential equation $x^{(\alpha)}(t)=g(x(t))$, $t\geq0$, where $\alpha\in(0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator…
We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange…
In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for diffusion-wave equation with the Caputo fractional derivative in a domain bounded by smooth curves. The uniqueness…
In this paper, for a degenerate higher order equation with a fractional derivative in the sense of Caputo, a nonlocal problem with conjugation conditions in a rectangular domain is studied. The solution is constructed in the form of a…
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution…