论文标题
基于新的 *r解决普通微分方程的视角
A novel *R-based perspective on solving ordinary differential equations
论文作者
论文摘要
实际数字是在大学中教授的，与我们的连续体的概念相对应，尽管超元数位于实际数字之间``之间''。数字$ x + dx $，其中$ dx $应该是无限的数字，而$ x $真实的数字无限量接近$ x $，但``无限''远离所有其他实数。类似地：如果$ f'（x_0）$和$ f（x_0）$给出了可区分函数$ f：\ mathbb {r} \ rightarrow \ rightarrow \ mathbb {r} $ at $ x_0 \ in \ in \ mathBb {r} $ $ x_0 $。这些要点似乎是``无限的''。这是在数值数学中求解微分方程的概念问题。在本文中，我们将提出一种数值算法来解决非常简单的初始值问题。但是，范式的更改是，我们不会``离开''点$ x_0 $。求解普通的微分方程就像搜索``食谱''$ f $。我们将从\ Mathbb {r} $中找到$ x \ $ x \的这些配方，而是从$ x_0 $的``monad''中的特殊关系中学习它们。
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$ real, is infinitesimally close to $x$ but ``infinitely'' far away from all other real numbers. Analogously: If $f'(x_0)$ and $f(x_0)$ are given for a differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ at $x_0\in\mathbb{R}$, we can not determine $f(x)$ at {\em any} point $x\in \mathbb{R}$ different from $x_0$. These points seem to be ``infinitely'' far away. That is one conceptual problem of solving differential equations in numerical mathematics. In this article, we will present a numerical algorithm to solve very simple initial value problems. However, the change of paradigm is, that we will not ``leave'' the point $x_0$. Solving ordinary differential equations is like searching for ``recipes'' $f$. Instead of trying to find these recipes for values $x\in\mathbb{R}$, we will learn them from special relations in the ``monad'' of $x_0$.
