相关论文: Construction of polynomial particular solutions of…

We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…

The main objective of this addendum to the mentioned article by Park is to provide some remarks on bifurcation theories for nonlinear partial differential equations (PDE) and their applications to fluid dynamics problems. We only wish to…

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and…

High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically…

The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear…

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the…

This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…

Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…

In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We apply this approach to…

The solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of…

We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure…

In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in $\mathbb{R}^d$, $d=2,3$. We consider the stochastic realizations using checkerboard configuration of the…

In a recent work [10], we have introduced a pressure-robust Hybrid High-Order method for the numerical solution of the incompressible Navier-Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error…

Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized…

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…

In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and…

Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…

Polynomial chaos expansions (PCE) allow us to propagate uncertainties in the coefficients of differential equations to the statistics of their solutions. Their main advantage is that they replace stochastic equations by systems of…

The Poisson structures for 3D systems possessing one constant of motion can always be constructed from the solution of a linear PDE. When two constants of the motion are available the problem reduces to a quadrature and the structure…