Related papers: Fast diffusion equation: uniqueness of solutions w…

We consider point sources in hyperbolic equations discretized by finite differences. If the source is stationary, appropriate source discretization has been shown to preserve the accuracy of the finite difference method. Moving point…

This article deals with time-fractional diffusion equations with time-dependent singular source term. Whenever the order of the time-fractional derivative is either multi-term, distributed or space-dependent, we prove that the system admits…

We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due…

Inverse problem to determine simultaneously a general space- and time-dependent source and an initial state in a fractional diffusion equation from an {\it a posteriori} measurement of the normal derivative of the state on a portion of a…

Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -\Delta_{p}u+|\nabla u|^q=0, \ \hbox{in} \…

The classical result by It\^o on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are…

We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions,…

In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.

Differential equations need boundary conditions (BC's) for their solution. It is commonly acknowledged that differential equations and BC's are representative of independent physical processes, and no correlations between them is required.…

Jeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive…

This article deals with an inverse problem of identifying the fractional order in the 1D time fractional diffusion equation (TFDE in short) using the measurement at one space-time point. Based on the expression of the solution to the…

In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. We use fixed point argument to give a new proof for the existence…

The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in…

We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansat\"atze we transform the PDE of diffusion to an ordinary…

This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data…

The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near…

It is known since Kellerer (1972) that for any process that is increasing for the convex order, or "peacock" as in Hirsch et al. 2011, there exist martingales with the same marginals laws. Nevertheless, there is no general constructive…

In the description of transport phenomena an important aspect represents the diffusion. In certain cases the diffusion may appear together with convection. In this paper we study the diffusion equation with the self similar Ansatz. With an…

This work contributes to an understanding of the domain size's effect on the existence and uniqueness of the linear convection--diffusion equation with integral-type boundary conditions, where boundary conditions depend non-locally on…