Numerical approximation to ODEs using the error functional

Abstract

Abstract. In this paper we present a new method for solving systems of ordinary nonlinear differential equations with initial conditions. The method is based on the transformation of the problem to an optimal control problem. We then solve it with a technique based on the use of an integral form of the Euler equation combined with the shooting method and the cyclic coordinate descent method. Our method substantially improves a previous approach that uses iterative dynamic programming to solve the associated optimal control problem. We consider the error functional instead of the classical global error, the error functional obtained by our method being lower than that obtained by classical methods. The method presented in this paper allows us to solve a wide range of nth order ordinary nonlinear differential equations with initial conditions.

Some Remarks on a Variational Method for Stiff Differential Equations

“…In this section, we compare our strategy using a fixed step implementation in the implicit Euler method, which is the simplest choice that we can consider, and the strategy used in [8]. We present the result of two examples studied in [8] (with a discretization of 1000 nodes), but similar conclusions are obtained for the other problems. At the end, we include an example to emphasize the additional advantages of our variable step implementation.…”

“…This initial article inspired [8] to set up an approximation scheme based on this optimization perspective. The authors of that contribution utilized this approach to reformulate it as an optimal control problem and then proposed an approximation technique for this related problem.…”

“…This note was motivated by [8], where inspired by our paper [7], the use of our error-functional strategy was considered but using a completely different way to derive the discretization. The particular transformation that the authors utilized to transform the variational problem into a control problem, in order to approximate the solution of the differential equation or system, seems a bit artificial, and nothing is missed if one deals directly with the variational problem to approximate the global minimizer numerically, i.e., the solution of the differential system (1).…”

“…We are interested in approximating the value x(1). See Table 1 for an explicit comparison of our iterative scheme versus the approximation suggested in [8]. Table 1.…”

“…Table 1. Comparison of our approach using the implicit Euler method and the approach proposed in [8].…”