Related papers: The Finite difference method for the Minkowski Cur…
In this paper, a mixed high order finite difference scheme-Pad\'{e} approximation method is applied to obtain numerical solution of the Riesz fractional advection-dispersion equation. This method is based on the high order finite difference…
We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other…
We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order…
A class of finite difference schemes for solving a fractional anti-diffusive equation, recently proposed by Andrew C. Fowler to describe the dynamics of dunes, is considered. Their linear stability is analyzed using the standard Von Neumann…
Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…
We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.
This paper presents a finite difference method combined with the Crank--Nicolson scheme of the Kuramoto--Sivashinsky equation defined on an expanding circle (\cite{KUY}), and the existence, uniqueness, and second-order error estimate of the…
We present a mimetic finite-difference approach for solving Maxwell's equations in one and two spatial dimensions. After introducing the governing equations and the classical Finite-Difference Time-Domain (FDTD) method, we describe mimetic…
Finite difference (FD) approximation is a classic approach to stochastic gradient estimation when only noisy function realizations are available. In this paper, we first provide a sample-driven method via the bootstrap technique to estimate…
This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the…
We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which…
This article aims to develop a direct numerical approach to solve the space-fractional partial differential equations (PDEs) based on a new differential quadrature (DQ) technique. The fractional derivatives are approximated by the weighted…
For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
This paper presents a new finite difference method, called {\varphi}-FD, inspired by the {\phi}-FEM approach for solving elliptic partial differential equations (PDEs) on general geometries. The proposed method uses Cartesian grids,…
We discuss $L_p$-estimates for finite difference schemes approximating parabolic, possibly degenerate, SPDEs, with initial conditions from $W^m_p$ and free terms taking values in $W^m_p.$ Consequences of these estimates include an…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
The goal of this paper is to clarify when the solutions to stochastic partial differential equations stay close to a given subset of the state space for starting points which are close as well. This includes results for deterministic…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…