An analytical approximation scheme to two-point boundary value problems of ordinary differential equations

Abstract

A new (algebraic) approximation scheme to find global solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose solutions are analytic near one of the boundary points. It is based on replacing the original ODE's by a sequence of auxiliary first order polynomial ODE's with constant coefficients. The coefficients in the auxiliary ODE's are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. To obtain the parameters of the global (connecting) solutions analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the "connecting parameters" for a number of nonlinear ODE's arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODE's coming from the exact renormalization group. The ground state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision.

Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111]

“…Finally, we mention that the HPM has recently proved successful for the treatment of other two-point nonlinear equations [31] of interest in some fields of physics [32][33][34].…”

Scalar field cosmologies with inverted potentials

“…κ so that (21) requires Φ ≥ Φ 0,min in that case. On the other hand, using (3) and (4), we obtain thatΦ = 0 is possible only for Φ ≥Φ when Z = 1 and for Φ ≤Φ when Z = −1 so that (19), (20) are immediately obtained. Remember that for Z = −1 we haveΦ > Φ 0,min .…”

“…For Z = 1, solutions withΦ 0 < 0, Φ 0,cr < Φ 0 < Φ max tend asymptotically to zero while solutions satisfyinġ Φ 0 < 0, Φ 0,min < Φ 0 < Φ 0,cr reach Φ max (and actually diverge) in a finite time and are therefore unviable as they leave the interval for which F > 0. There is a critical value Φ 0,cr separating the two behaviours [19] and solutions starting from Φ 0,cr withΦ 0 < 0 tend tõ Φ. All solutions withΦ 0 > 0 will diverge in the future in a finite time.…”

Bouncing universes in scalar-tensor gravity models admitting negative potentials

“…So this solution which tends to the saddle point is the separatrix which separates the two types of behaviours. Hence for given parameters κ, Λ and c satisfying (19), the two types of behaviours are possible provided the existence of the separatrix.…”