We consider a fractional diffusion equation (FDE) = a D u a t u C t xx ( ) with an undetermined time-dependent diffusion coefficient a(t). Firstly, for the direct problem part, we establish the existence, uniqueness and some regularity properties of the weak solution for this FDE with a fixed a t . ( ) Secondly, for the inverse problem part, in order to recover a t , ( ) we introduce an operato...

We study the inverse boundary value problem on determining a space-dependent component in right-hand side of semilinear time fractional diffusion-wave equation. find sufficient conditions for time-local uniqueness solution under time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Omega\subset \Bbb R^n\] where $u$ is unknown first such equation, $...

In this paper, we consider the inverse nodal problem for conformable fractional diffusion operator with parameter-dependent Bitsadze–Samarskii type nonlocal boundary condition. We obtain asymptotics eigenvalues, eigenfunctions, and zeros of eigenfunctions (called points or nodes) considered operator, provide a constructive procedure solving problem, i.e., reconstruct potential functions p(x) q(...

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as wel...

In many applications, such as the heat conduction and hydrology, there is a need to recover the (possibly discontinuous) diffusion coefficient a from boundary measurements of solutions of a parabolic equation. The complete inverse problem is ill posed and nonlinear, so numerical solution is quite difficult, and we linearize the problem around constant a. We study and solve numerically the linea...