A “Discretization” Technique for the Solution of ODEs II

Abstract

A functional analytic technique was recently presented for finding discrete equivalent counterparts of initial value problems of ODEs and obtaining their real analytic solutions. In the current paper, this technique is extended to boundary value problems of ODEs and to the complex solutions of ODEs. In order to demonstrate this technique, it is applied to the classic Blasius problem of fluid mechanics. Apart from its real solution, its complex solution is also studied. The obtained results indicate that the complex Blasius function exhibits an oscillatory behavior and strengthen a conjecture regarding its singularities in the complex plane.

On the Logistic Equation in the Complex Plane

“…In this section, the method of [15] and [16] is described by applying it to the logistic differential equation (2.3) for a, , , t , y(t ) complex, through Steps 1-6. Steps 1-4 describe in a more algorithmic way the study performed in Section 2.2 and thus they will be given without many details.…”

“…"Numerical" implementation: In order to proceed to the numerical implementation of the method and the computation of the solution of (2.3) the procedure followed in [15,16] is considered. The first step is to determine T as described above and then, determine the coefficients Y n .…”

“…In [15,16], a nonstandard way was proposed for the "numerical" solution of an ordinary differential equation (ODE) accompanied by initial or boundary conditions in the real or complex plane. This method is based on a functional analytic technique, which answers questions regarding the existence and uniqueness of solutions of ODEs and ordinary difference equations (O Es) in the Hilbert spaces (1.4) where = z ∈ : |z| < 1 and…”

“…In this article, the method of [15] and [16] will be applied to (1.1), so as to obtain unique complex analytic solutions of (1.1) of the form (1.8), where the coefficients Y n will now satisfy (1.9) with a, , , Y n complex, as will be shown in Section 2.3. By finding regions in where (1.1) has a unique analytic solution, it is possible to detect regions of where possible singularities of its solution occur and the first of these singularities can be numerically located, as it will be demonstrated in Section 3.…”

“…Mathematical curiosity is one of them! This is one of the reasons for studying the complex solutions of the well known, in fluid mechanics, Blasius problem in [18] and [16]. Furthermore, the investigation of physical problems for complex values of physical parameters can in many cases provide a further insight of the properties of their solutions as in [19][20][21].…”