Strict LpSolutions for Nonautonomous Fractional Evolution Equations

MSC 2010: 26A33, 34A08, 34K3

An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid

We study the Rayleigh-Stokes problem for a generalized second-grade fluid which involves a Riemann-Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data $v$, including $v\in L^2(\Omega)$. A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.Comment: 23 pp, 4 figures. The error analysis of the fully discrete scheme is shortene

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)”, contract No DO1–205/23.11.2018, financed by the Ministry of Education and Science in Bulgaria

Принцип за субординация на обобщени дробни еволюционни уравнения

ИМИ-БАН, 15.11.2022 г., присъждане на научна степен "доктор на науките" на Емилия Григорова Бажлекова. [Bazhlekova Emilia Grigorova; Бажлекова Емилия Григорова

Peristaltic transport of viscoelastic bio-fluids with fractional derivative models

Peristaltic flow of viscoelastic fluid through a uniform channel is considered under the assumptions of long wavelength and low Reynolds number. The fractional Oldroyd-B constitutive viscoelastic law is employed. Based on models for peristaltic viscoelastic flows given in a series of papers by Tripathi et al. (e.g. Appl Math Comput. 215 (2010) 3645–3654; Math Biosci. 233 (2011) 90–97) we present a detailed analytical and numerical study of the evolution in time of the pressure gradient across one wavelength. An analytical expression for the pressure gradient is obtained in terms of Mittag-Leffler functions and its behavior is analyzed. For numerical computation the fractional Adams method is used. The influence of the different material parameters is discussed, as well as constraints on the parameters under which the model is physically meaningful

A Compact Alternating Direction Implicit Scheme for Two-Dimensional Fractional Oldroyd-B Fluids

[Vasileva Daniela; Василева Даниела]; [Bazhlekov Ivan; Бажлеков Иван]; [Bazhlekova Emilia; Бажлекова Емилия]The two-dimensional Rayleigh-Stokes problem for a generalized Oldroyd-B fluid is considered in the present work. The fractional time derivatives are discretized using L1 and L2 approximations. A fourth order compact approximation is implemented for the space derivatives and two variants of an alternating direction implicit finite difference scheme are numerically investigated. 2010 Mathematics Subject Classification: 26A33, 35R11, 65M06, 65M22, 74D05

Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem

[Bazhlekova Emilia; Бажлекова Емилия]The final value problem for the heat equation is known to be ill-posed. To deal with this, in the method of quasi-reversibility (QR), the equation or the final value condition is perturbed to form an approximate well-posed problem, depending on a small parameter ε. In this work, four known quasi-reversibility techniques for the backward heat problem are considered and the corresponding regularizing problems are treated using the convolutional calculus approach developed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht, 1990). For every regularizing problem, applying an appropriate bivariate convolutional calculus, a Duhamel-type representation of the solution is obtained. It is in the form of a convolution product of a special solution of the problem and the given final value function. A non-classical convolution with respect to the space variable is used. Based on the obtained representations, numerical experiments are performed for some test problems. 2010 Mathematics Subject Classification: 35C10, 35R30, 44A35, 44A40