Related papers: Complex singularities and PDEs
The so-called 'direct' approach to separation of variables in linear PDEs is applied to the hydrodynamic stability problem. Calculations are made for the complete linear stability equations in cylindrical coordinates. Several classes of the…
We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is…
In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong…
We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schr\"odinger-type equations. The approach is based on the observation that generic solutions of…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Pad\'e summation, self-similar factor…
While real-valued solutions of the Korteweg--de Vries (KdV) equation have been studied extensively over the past 50 years, much less attention has been devoted to solution behaviour in the complex plane. Here we consider the analytic…
We evaluate an efficient overset grid method for two-dimensional and three-dimensional particulate flows for small numbers of particles at finite Reynolds number. The rigid particles are discretised using moving overset grids overlaid on a…
The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
We extensively study the numerical accuracy of the well-known time splitting Fourier spectral method for the approximation of singular solutions of the Gross-Pitaevskii equation. In particular, we explore its capability of preserving a…
Asymptotic properties of solutions of odd-order nonlinear dispersion equations are studied. The global in time similarity solutions, which lead to eigenfunctions of the rescaled ODEs, are constructed.
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic…
We propose a continuous data assimilation (CDA) method to address the uniqueness problem for steady Navier-Stokes equations(NSE). The CDA method incorporates spatial observations into the NSE, and we prove that with sufficient observations,…
We list all connected components of sets of non-discriminant functions near all {\em parabolic} function singularities (which are the second most important family of singularity classes of smooth functions after {\em simple} singularities).…
We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…
We propose and study a temporal, and spatio-temporal discretisation of the 2D stochastic Navier--Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the…
We present an efficient algorithm for computing the closest singular configuration to each non-singular pose of a 3-RPR planar manipulator performing a 1-parametric motion. By considering a 3-RPR manipulator as a planar framework, one can…