Numerical approximation to ODEs using the error functional (Q2846854)

This paper is concerned with the global approximation of solutions of initial value problems (IVPs) for ordinary differential equations. For a scalar IVP : \( \dot{x} (t) = f(t, x(t))\), \( t \in [a,b]\), \( x(a)= x_a\), the authors start formulating the equivalent minimization problem: Find \( x = x(t)\), \( t \in [a,b]\), so that the functional \( E(x, \dot{x}) \equiv \int_a^b | \dot{x} (t) - f(t, x(t)) |^2 dt \) subject to \( x(a) = x_a\) is minimized. Next, this problem is stated as the optimal control problem \( \min \{ E(x,u) ;~ \dot{x} (t) = u(t),~ x(a)= x_a \} \) that is solved by using Pontryagin's maximum principle. In this way in the final form, the authors are faced to solve a boundary value problem and in practical computation this is carried out by approximating the solution in a uniform grid by means of Euler's polygonals that minimize the error integral functional integral \( E( x , \dot{x} ) \). Hence, in contrast with standard step by step methods that attempt to minimize the global error at some grid points in the integration interval, here the goal is to minimize the error integral functional \( E( x , \dot{x} ) \). This approach is stated also for systems of equations and some comments are given for solving the nonlinear equations that arise in the optimization process. Finally, the results of a number of numerical examples with simple test problems are presented to show that the proposed method is able to get better accuracy than some standard methods but usually the price is a higher computational cost.