arXiv subjects

Nataliya Vasylyeva

Publications and source records attributed to Nataliya Vasylyeva.

11 recordsLinked to original sources

arXiv2026

Some Unexplored Topics in The Reconstruction of Scalar Parameters of Subdiffusion

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.

math.AP↗

arXiv2026

On the explicit formula linking a function to the order of its fractional derivative

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.

math.AP↗

arXiv2025

Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation

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.

math.AP↗

arXiv2024

On a local solvability of the contact Muskat problem

In the paper, we discuss the two-dimensional contact Muskat problem with zero surface tension of a free boundary. The initial shape of the unknown interface is a smooth simple curve which forms acute corners $δ_{0}$ and $δ_{1}$ with fixed boundaries. Under suitable assumptions on the given data, the one-to-one local classical solvability of this problem is proved. We also describe the sufficient conditions on the data in the model which provide the existence of the "waiting time" phenomenon.

math.AP↗

arXiv2024

Analysis of a Radiotherapy Model for Brain Tumors

In this work, we focus on the analytical and numerical study of a mathematical model for brain tumors with radiotherapy influence. Under certain assumptions on the given data in the model, we prove existence and uniqueness of a weak nonnegative (biological relevant) solution. Then, assuming only more regular initial data, we obtain the extra regularity of this solution. Besides, we analyze the optimal control of the advection coefficient responding for the radiotherapy effect on the tumor cell population. Finally, we provide numerical illustration to all obtained analytical results.

math.AP↗

arXiv2024

Longtime behavior of semilinear multi-term fractional in time diffusion

In the paper, the initial-boundary value problems to a semilinear integro-differential equation with multi-term fractional Caputo derivatives are analyzed. A particular case of this equation models oxygen diffusion through capillaries. Under proper assumptions on the coefficients and a nonlinearity, the longtime behavior (as $t\to+\infty$) of a solution is discussed. In particular, the existence of absorbing sets in suitable functional spaces is established.

math.AP↗

arXiv2023

Initial-boundary value problems to semilinear multi-term fractional differential equations

For $ν,ν_i,μ_j\in(0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $Ω=(a,b)$ in the unknown $u=u(x,t)$ \[ \mathbf{D}_{t}^ν(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{ν_{i}}(\varrho_{i}u) -\sum_{j=1}^{N}\mathbf{D}_{t}^{μ_{j}}(γ_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t), \] where $\mathbf{D}_{t}^ν,\mathbf{D}_{t}^{ν_{i}}, \mathbf{D}_{t}^{μ_{j}}$ are Caputo fractional derivatives, $\varrho_0=\varrho_0(t)>0,$ $\varrho_{i}=\varrho_{i}(t)$, $γ_{j}=γ_{j}(t)$, $\mathcal{L}_{k}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $f$ and orders $ν,ν_i,μ_j$, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.

math.AP↗

arXiv2022

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

For $ν\in[0,1]$ and a complex parameter $σ,$ $Re\, σ>0,$ we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane $z\in\mathbb{C}$: \[ (a_{1}σ+a_{2}σ^ν)\mathcal{Y}(z+β,σ)-Ω(z)\mathcal{Y}(z,σ)=\mathbb F(z,σ), \quadβ\in\mathbb{R},\, β\neq 0, \] where $Ω(z)$ and $\mathbb{F}(z)$ are given complex functions, while $a_{1}$ and $a_{2}$ are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as $|z|\to +\infty$. Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed.

math.AP↗

arXiv2022

Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

For $0<ν_2<ν_1\leq 1$, we analyze a linear integro-differential equation on the space-time cylinder $Ω\times(0,T)$ in the unknown $u=u(x,t)$ $$\mathbf{D}_{t}^{ν_1}(\varrho_{1}u)-\mathbf{D}_{t}^{ν_2}(\varrho_2 u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u =f$$ where $\mathbf{D}_{t}^{ν_i}$ are the Caputo fractional derivatives, $\varrho_i=\varrho_i(x,t)$ with $\varrho_1\geq μ_0>0$, $\mathcal{L}_{i}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel, and $f$ is an external force. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under suitable conditions on the given data, the global classical solvability of the associated initial-boundary value problems is addressed. To this end, a special technique is needed, adapting the concept of a regularizer from the theory of parabolic equations. This allows us to remove the usual assumption about the nonnegativity of the kernel representing fractional derivatives. The problem is also investigated from the numerical point of view.

math.AP↗

arXiv2022

Qualitative analysis of solutions for a degenerate PDE model of epidemic dynamics

Compartmental models are widely used in mathematical epidemiology to describe dynamics of infection disease. A new SIS-PDE model, recently derived by Chalub and Souza, is based on a diffusion-drift approximation of probability density in a well-known discrete - time Markov chain SIS-DTMC model. This new SIS-PDE model is conservative due to degeneracy of the diffusion term at the origin. We analyze a class of degenerate PDE models and obtain sufficient conditions for existence of classical solutions with certain regularity properties. Also, we show that under some additional assumptions about coefficients and initial data classical solutions vanish at the origin at any finite time. Vanishing at the origin of solutions is consistent with the conservation property of the model. The main results of this article are: sufficient conditions for existence of weak solutions, analysis of their asymptotic behavior at the origin, and the proof of existence of weak solutions convergent to Dirac delta function. Moreover, we study long-time behavior of solutions and confirm our analysis by numerical computations.

math.AP↗

arXiv2020

Asymptotic analysis of a contact Hele-Shaw problem in a thin domain

We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain $Ω^{\varepsilon}(t).$ Under suitable conditions on the given data, the one-valued local classical solvability of the problem for each fixed value of the parameter $\varepsilon$ is proved. Using the multiscale analysis, we study the asymptotic behavior of this problem as $\varepsilon \to 0,$ i.e., when the thin domain $Ω^{\varepsilon}(t)$ is shrunk into the interval $(0, l).$ Namely, we find exact representation of the free boundary for $t\in[0,T],$ derive the corresponding limit problem $(\varepsilon= 0),$ define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify this approach. We also establish the preserving geometry of the free boundary near corner points for $t\in[0,T]$ under assumption that free and fixed boundaries form right angles at the initial time $t=0$.

math.AP↗