Miglena Nikolaeva Koleva

Many important physical phenomena can be described by parabolic interface problems. An example is provided by a two-layer magnetic system symmetrically distributed over two disconnected domains. In this work, we consider an inverse problem for identifying a space-dependent diffusion coefficient in a parabolic equation defined on disjoint domains, based on final-time measurements. First, we discuss the well-posedness of the corresponding direct problem. Then, we develop an iterative method for solving the inverse problem, which is reformulated as a sequence of direct problems. A numerical implementation based on the finite difference method is presented, and results from numerical test examples are discussed.

This paper focuses on the identification of boundary conditions in tempered fractional ordinary differential equations interface problem on disjoint intervals based on point observations (measurements). First, we discuss some properties of tempered fractional calculus, including fractional integration by parts in appropriate Sobolev spaces. Then, after formulating the direct and inverse problems, we establish the well-posedness of a tempered fractional boundary-value problem on disjoint intervals. Two numerical methods for solving the inverse problem of determining the external boundary conditions are proposed, and their correctness is studied. Finally, numerical experiments are presented to demonstrate the efficiency of the proposed approaches.

Mathematical modeling of heat and mass transfer processes in porous media using fractional derivative equations is of great practical importance. Within the framework of such models, obtaining analytical solutions to the corresponding initial–boundary value problems is generally difficult. In this work, we numerically investigate quasilinear parabolic problems involving Caputo time-fractional derivatives. First, the well-posedness and existence of weak solutions are discussed. Then, we construct and implement a finite-difference scheme that is fourth-order accurate in space and second-order accurate in time. Convergence in the maximum norm is proven. Numerical experiments confirm the accuracy and efficiency of the proposed approach.

In the numerical simulation of composite material models, it is often necessary to recover boundary values of the solution to parabolic problems from integral constraints. In this work, we consider an inverse problem of determining a Dirichlet boundary condition for a heat equation with multiple interfaces and integral overspecification on a part of the spatial domain. After establishing the well-posedness of the direct problem, we propose an efficient numerical method for identifying an unknown Dirichlet boundary condition. The method decomposes the global inverse problem into a sequence of local subproblems, solved independently within each layer, including the accurate reconstruction of the solution at the interfaces. The approach relies solely on explicit schemes, employing an unconditionally stable Saulyev-type discretization and a novel interface treatment that avoids matrix inversion. Results from numerical experiments are presented and discussed.