Related papers: A splitting method for nonlinear diffusions with n…
In ecological studies of pattern formation, models of the competitive-diffusion type are generally singularly perturbed, and the numerical approximation of such models is challenging. In this paper, we present finite element discretization…
We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear…
The two-dimensional Terry-Horton equation is shown to exhibit the Dimits shift when suitably modified to capture both the nonlinear enhancement of zonal/drift-wave interactions and the existence of residual Rosenbluth-Hinton states. This…
We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting…
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity…
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
In this paper we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains. The existence of an orthonormal basis of eigenfunctions is established making use of weak T-coercivity theory. All…
Let $x$ denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of…
We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives,…
We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order…
We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We…
This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient…
A scheme is developed for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from time-series data. The scheme does not require to specify parametric forms for the drift and diffusion…
We consider a non-linear variant of the transport-diffusion osmosis model for solving a variety of imaging problems such as shadow/soft-light removal and compact data representation. The non-linear behaviour is encoded in terms of a general…
We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one…
In this paper, we derive two bound-preserving and mass-conserving schemes based on the fractional-step method and high-order compact (HOC) finite difference method for nonlinear convection-dominated diffusion equations. We split the…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial…
We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data…