Related papers: On numerical methods and error estimates for degen…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the…
A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection--diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal…
We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a…
We present a new third-order, semi-discrete, central method for approximating solutions to multi-dimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension…
We propose two closely--related Lagrangian numerical methods for the simulation of physical processes involving advection, reaction and diffusion. The methods are intended to be used in settings where the flow is nearly incompressible and…
In this paper we study some cases of time-fractional nonlinear dispersive equations (NDEs) involving Caputo derivatives, by means of the invariant subspace method. This method allows to find exact solutions to nonlinear time-fractional…
We derive optimal $L^2$-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order $\alpha\in (0,1)$, for cases with smooth and nonsmooth initial data. A general framework is…
We presented a general approach for obtaining the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and Zubarev's nonequilibrium…
Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…
We study the systematic numerical approximation of Maxwell's equations in dispersive media. Two discretization strategies are considered, one based on a traditional leapfrog time integration method and the other based on convolution…
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques,…
Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients.
In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on…
The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional…
In this paper, using a pseudospectral approach, we develop operational matrices based on the shifted Chebyshev polynomials to approximate numerically Caputo fractional derivatives and Riemann-Liouville fractional integrals. In order to make…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…