论文标题
使用共轭解决普通微分方程的逆问题
Solving the inverse problem for an ordinary differential equation using conjugation
论文作者
论文摘要
我们考虑普通微分方程（ODE）的以下反问题：给定一组数据点$ p = \ {（t_i，x_i），\; i = 1，\ dots，n \} $，找到一个ode $ x^\ prime（t）= v（x）$，该$允许解决方案$ x（t）$，以便尽可能接近$ x_i \ oft x（t_i）$。提出方法的关键是从给定数据集中找到递归或离散传播函数$ d（x）$的近似值。之后，我们使用Schröder方程定义的共轭图和相关朱莉娅方程的解决方案来确定字段$ V（x）$。此外，我们的方法还适用于从多组数据点确定ODE的反问题。我们还研究了回收的字段$ v（x）$的存在，独特性，稳定性和其他特性。最后，我们提出了几种数值方法，用于近似字段$ V（x）$，并提供了这些方法应用的一些说明性示例。
We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points $P=\{(t_i,x_i),\; i=1,\dots,N\}$, find an ODE $x^\prime(t) = v (x)$ that admits a solution $x(t)$ such that $x_i \approx x(t_i)$ as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function $D(x)$ from the given data set. Afterwards, we determine the field $v(x)$, using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Moreover, our approach also works for the inverse problems where one has to determine an ODE from multiple sets of data points. We also study existence, uniqueness, stability and other properties of the recovered field $v(x)$. Finally, we present several numerical methods for the approximation of the field $v(x)$ and provide some illustrative examples of the application of these methods.
