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This paper is devoted to the study of the inverse problem of determining the right-hand side of the subdiffusion equation with the Caputo derivative with respect to time. In our case, the inverse problem consists in restoring the…
In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…
The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and…
We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state…
We study a one-dimensional nonlinear hyperbolic-parabolic initial boundary value problem occurring in the theory of thermoelasticity. We prove existence and uniqueness of the local-in-time strong solution. Also, some global-in-time weak…
An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a…
In this paper we prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in $\mathbb{R}^n$, from a knowledge, in a finite time observation, of…
An approximately globally convergent numerical method for a 3d Coefficient Inverse Problem for a hyperbolic equation with backscattering data is presented. A new approximate mathematical model is presented. An approximation is used only on…
This paper is concerned exclusively with axisymmetric spacetimes. We want to develop reductions of Einstein's equations which are suitable for numerical evolutions. We first make a Kaluza-Klein type dimensional reduction followed by an ADM…
Recently, results regarding the Inverse Design problem for Conservation Laws and Hamilton-Jacobi equations with space-dependent convex fluxes were obtaine. More precisely, characterizations of attainable sets and the set of initialdata…
We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative…
We present a family of integral equation-based solvers for the heat equation, reaction-diffusion systems, the unsteady Stokes equation and the incompressible Navier-Stokes equations in two space dimensions. Our emphasis is on the…
This paper presents a backstepping approach for the boundary control of first-order hyperbolic equations with spatially varying coefficients posed on domains of arbitrary dimension. The method is based on a change of variables induced by…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…
We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.
A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave…
We consider a partial data inverse problem for a time-dependent convection-diffusion equation on an admissible manifold. We prove that the time-dependent convection term and time-dependent density can be recovered uniquely modulo a known…
This article proves the uniqueness for two kinds of inverse problems of identifying fractional orders in diffusion equations with multiple time-fractional derivatives by pointwise observation. By means of eigenfunction expansion and Laplace…