Nataliya Vasylyeva

Andrii Hulianytskyi, Sergei Pereverzyev, Sergii Siryk et al.

In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, $\mathbf{D}_t$. To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation $\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=g(x,t),$ where $\mathcal{L}_{i}$ are the second order elliptic operators with time-dependent coefficients, $\mathcal{K}$ is a summable memory kernel, and $g$ is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.

Vasyl Semenov, Nataliya Vasylyeva

In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $ν_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $ν_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.

Sergii V. Siryk, Lidiia Tereshchenko, Nataliya Vasylyeva

In the paper, we discuss the reconstruction of scalar parameters in a linear diffusion equation with fractional in time differential operators and with additional nonlocal (convolution) terms, which incorporate memory effects in models. Although, under suitable assumptions on the data, inverse problems associated with recovery of these parameters are nowadays well understood, several important questions related with numerical reconstructions of these parameters via a nonlocal observation in a small time interval have not yet been analyzed. This paper aims to provide some answers.