A “discretization” technique for the solution of ODEs

Abstract

A functional-analytic technique was developed in the past for the establishment of unique solutions of ODEs in H 2 (D) and H 1 (D) and of difference equations in 2 and 1 . This technique is based on two isomorphisms between the involved spaces. In this paper, the two isomorphisms are combined in order to find discrete equivalent counterparts of ODEs, so as to obtain eventually the solution of the ODEs under consideration. As an application, the Duffing equation and the Lorenz system are studied. The results are compared with numerical ones obtained using the 4th order Runge-Kutta method. The advantages of the present method are that, it is accurate, the only errors involved are the round-off errors, it does not depend on the grid used and the obtained solution is proved to be unique.

A “Discretization” Technique for the Solution of ODEs II

“…As already mentioned in the introduction, the technique that will be used in the current paper for the study of the Blasius problem is based on a functional analytic method introduced in [1]. More precisely, the main idea is to transform the ordinary differential equation under consideration, i.e., the Blasius equation, into an equivalent operator equation in an abstract Banach space and from this to deduce an equivalent difference equation, which is the "numerical scheme" used in order to obtain an exact solution of the Blasius problem.…”

“…In [1], a functional-analytic technique was introduced for the discretization of initial value problems of (system of) nonlinear ordinary differential equations. This technique is based on the equivalent transformation of (system of) ordinary differential equation(s), to a (system of) difference equation(s), through an operator equation (or system of operator equations), via specific isomorphisms.…”

“…This technique is based on the equivalent transformation of (system of) ordinary differential equation(s), to a (system of) difference equation(s), through an operator equation (or system of operator equations), via specific isomorphisms. As an application, the real analytic solutions of the Duffing oscillator equation and the results were compared with numerical ones obtained using the classic fourth order Runge-Kutta method, indicating that the method in [1] is better than the Runge-Kutta method, with respect to the accuracy and the CPU time required. Moreover, the advantages of this method are that (i) a discrete equivalent and not a discrete analogue of the (system of) differential equation(s) is obtained, which makes the method very accurate as the only errors involved are the round-off errors; (ii) it is independent of the grid used; and (iii) it is very quick.…”

“…The principal aim of the current paper is to extend the method of [1], to (i) the study of boundary value problems of ordinary differential equations; and (ii) the evaluation of complex solutions of ordinary differential equations.…”

“…The first is accomplished by combining the technique in [1] with a classic "shooting" method and solving, instead of the boundary value problem under consideration, a "sequence" of associated initial value problems. The second is accomplished by making the simple observation that the coefficients f n of (1.1) can be written in the form f n = u n + iv n and the independent variable z (which is now a complex variable in the disk |z| < T ) in the polar form z = re i .…”