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We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite…
This work considers to numerically solve a subdiffusion equation involving constant time delay $\tau$ and Riemann-Liouville fractional derivative. First, a fully discrete finite element scheme is developed for the considered problem under…
An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability…
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as…
This paper summarizes a research program that has been underway for a decade. The objective is to find a fast and accurate scheme for solving quantum problems which does not involve a Monte Carlo algorithm. We use an alternative strategy…
We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the…
As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly employed in the related…
In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential…
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…
A random walk-based method is proposed to efficiently compute the solution of a large class of fractional in time linear systems of differential equations (linear F-ODE systems), along with the derivatives with respect to the system…
A fully implicit numerical scheme is established for solving the time fractional Swift-Hohenberg (TFSH) equation with a Caputo time derivative of order $\alpha\in(0,1)$. The variable-step L1 formula and the finite difference method are…
Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn…
We introduce a two-phase approximation method designed to resolve singularities in three-dimensional harmonic Dirichlet problems. The approach utilizes the classical Green's function representation, decomposing the function into its…
Fractures are normally present in the underground and are, for some physical processes, of paramount importance. Their accurate description is fundamental to obtain reliable numerical outcomes useful, e.g., for energy management. Depending…
We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original…
A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and…
This work uses a linear relaxation method to develop efficient numerical schemes for the time-fractional Allen-Cahn and Cahn-Hilliard equations. The L1+-CN formula is used to discretize the fractional derivative, and an auxiliary variable…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…