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The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing…
This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or…
The time-fractional porous medium equation is an important model of many hydrological, physical, and chemical flows. We study its self-similar solutions, which make up the profiles of many important experimentally measured situations. We…
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the non-homogeneous diffusion coefficient…
This paper explores the forward and inverse problems for a fractional subdiffusion equation characterized by time-dependent diffusion and reaction coefficients. Initially, the forward problem is examined, and its unique solvability is…
We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular…
The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a…
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of…
This paper investigates the initial value problem for a system of one-dimensional fourth-order dispersive partial differential-integral equations with nonlinearity involving derivatives up to second order. Examples of the system arise in…
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution…
We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and…
In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second order systems of wave equations be strongly well-posed in a generalized sense. The applications included the…
Difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $\alpha$ and $\beta$ are considered. By the method of energy inequalities, for the solution…
In this work, we study an inverse problem of recovering a space-time dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian-Caputo fractional derivative…
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\beta \in (0,1)$. The fundamental solution for the Cauchy problem is…
We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence,…
We introduce a very weak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time…
This paper is devoted to the investigation of the backward problem for a multi-term time-fractional diffusion equation. Backward problems for fractional diffusion equations are typically studied using regularization methods due to their…
In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the…
We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to…