Related papers: Unidirectional evolution equations of diffusion ty…
We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such…
Mathematical modeling of many physical processes such as diffusion, viscosity of fluids and combustion involves differential equations with small coefficients of higher derivatives. These may be small diffusion coefficients for modeling the…
Equations for dislocation evolution bridge the gap between dislocation properties and continuum descriptions of plastic behavior of crystalline materials. Computer simulations can help us verify these evolution equations and find their…
In the present article, we study the diffusion equations with fractional time derivatives. The aim of this paper is to investigate the best possible regularity for the initial value/boundary value problems with non-homogeneous Dirichlet…
There has been considerable recent study in "sub-diffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. There are many ways to look at this newer approach and one…
The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed. All the elementary properties of the classical Laplace transform are extended to the conformable fractional transform, and using…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave…
We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite…
A system of a first order history-dependent evolutionary variational-hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem…
We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1…
This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells…
We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such…
We study evolution equations of drift-diffusion type when various parameters are random. Motivated by applications in pedestrian dynamics, we focus on the case when the total mass is, due to boundary or reaction terms, not conserved. After…
We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal…
Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of energy inequalities is applicable to obtaining a priori…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and…
This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the…
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency…
We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale…