Solving the inverse problem for an ordinary differential equation using conjugation

Alfaro Vigo, D. G, Alvarez, A. C, Chapiro, G., Garcia-Mokina, G., Moreira, C. G. T. A

Published 2020-03-29Version 1

We consider the following inverse problem for an ordinary differential equations (ODE): given a set of data points $(t_i,x_i)$, $i=1,\cdots,N$, find an ODE $x^\prime(t) = v (x(t))$ that admits a solution $x(t)$ such that $x_i = x(t_i)$. To determine the field $v(x)$, we use the conjugate map defined by Schr\"{o}der equation and the solution of a related Julia equation. We also study existence, uniqueness, stability and other properties of this solution. 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.