Related papers: Complex singularities and PDEs
In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously…
We present a review of the discrete dipole approximation (DDA), which is a general method to simulate light scattering by arbitrarily shaped particles. We put the method in historical context and discuss recent developments, taking the…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…
This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network…
Very singular self-similar solutions of semilinear odd-order PDEs are studied on the basis of a Hermitian-type spectral theory for linear rescaled odd-order operators.
A simplified version of Hirota's method for the computation of solitary waves and solitons of nonlinear PDEs is presented. A change of dependent variable transforms the PDE into an equation that is homogeneous of degree. Solitons are then…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
In this paper, we study the singularities of Feynman integrals using homological techniques. We analyse the Feynman integrals by compactifying the integration domain as well as the ambient space by embedding them in higher-dimensional…
A method for computing singular or nearly singular integrals on closed surfaces was presented by J. T. Beale, W. Ying, and J. R. Wilson [Comm. Comput. Phys. 20 (2016), 733--753, arXiv:1508.00265] and applied to single and double layer…
By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of…
We analysis some singular partial differential equations systems(PDAEs) with boundary conditions in high dimension bounded domain with sufficiently smooth boundary. With the eigenvalue theory of PDE the systems initially is formulated as an…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental…
Discrete dynamical systems over finite fields are investigated and their integrability is discussed. In particular, the discrete Painlev\'{e} equations and the discrete KdV equation are defined over finite fields and their special solutions…
This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…
We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with…