Each example applies a technique that reduces N-dimensional differential systems into 1D separable parts, then solves them using exponential product formulas (Lie-Trotter, Strang, or Higher order ...

We provide a new analytical approach to operator splitting for equations of the type u t = Au + uu x where A is a linear differential operator such that the equation is well-posed. Particular examples ...

Pseudodifferential operators serve as a pivotal extension of classical differential operators by incorporating non-local features through their symbols. These operators are fundamental in the analysis ...

Neural networks have been widely used to solve partial differential equations (PDEs) in different fields, such as biology, physics, and materials science. Although current research focuses on PDEs ...

ABSTRACT: The operator T from a domain D into the space of measurable functions is called a nonanticipating (causal) operator if the past information is independent from the future outputs. We will ...

ABSTRACT: The two-dimensional nonlinear shallow water equations in the presence of Coriolis force and bottom topography are solved numerically using the fractional steps method. The fractional steps ...

Recently, deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited ...

Abstract: Neural operators are a class of neural networks to learn mappings between infinite-dimensional function spaces, and recent studies have shown that using neural operators to solve partial ...