相关论文: Extending nonstandard finite difference schemes ru…
We consider a dynamical system, defined by a system of autonomous differential equations, on $\Omega\subset\mathbb{R}^n$. By using Mickens' rule on the nonlocal approximation of nonlinear terms, we construct an implicit Nonstandard Finite…
In this work, a class of continuous-time autonomous dynamical systems describing many important phenomena and processes arising in real-world applications is considered. We apply the nonstandard finite difference (NSFD) methodology proposed…
In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…
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 study the construction of a non-standard finite differences numerical scheme for a general class of two dimensional differential equations including several models in population dynamics using the idea of non-local approximation…
In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
Standard finite difference (SFD) schemes often suffer from limited stability regions, especially when applied in explicit setup to partial differential equations. To address this challenge, this study investigates the efficacy of…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
We construct a nonstandard finite difference (NSFD) scheme for an SIRS mathematical model of respiratory virus transmission. This discretization is in full compliance with the NSFD methodology as formulated by R. E. Mickens. By use of an…
This paper proposes a novel Generalized Non-Standard Finite Difference (GNSFD) scheme for the numerical solution of a class of fractional partial differential equations (FrPDEs). The formulation of the method is grounded in optimization and…
In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under dissipative conditions, numerous previous works have established rigorous…
Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
The solution of large systems of nonlinear differential equations is needed for many applications in science and engineering. In this study, we present three main improvements to existing quantum algorithms based on the Carleman…
A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order…