Solving Equations with Physical Understanding

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

“…Similarly, the same functions are approximated in [8]. However, strictly speaking, it can be seen that neither equation (1), nor its associated planar system ẋ = y,…”

“…For system (2) this is so, because it has no critical points and it is well-known ( [9]) that periodic orbits of planar autonomous systems must surround a critical point. For equation (1) it is not difficult to see that any periodic solution x(t) must vanish for some t * ∈ R, that is x(t * ) = 0. Then lim t→t * ẍ(t) = ∞ and as a consequence this equation cannot have C 2 -periodic solutions.…”

“…More concretely, studying a certain electron beam injected into a plasma tube, where the magnetic field is cylindrical and increases towards the axis in inverse proportion to the radius, see [1,8]. In [11] and [13,Sec.…”

“…The nonlinear differential equation xẍ + 1 = 0 (1) appears in the modeling of certain phenomena in plasma physics. More concretely, studying a certain electron beam injected into a plasma tube, where the magnetic field is cylindrical and increases towards the axis in inverse proportion to the radius, see [1,8].…”

An improved CAD procedure for oscillator circuit design using fast-approximation formulas describing the voltage-dependent non-linear characteristics of the intrinsic equivalent circuit elements of the active devices

“…At first, the measured data is plotted in order to clarify the course of the context. Then, the fundamental structures of the approximation formulas have to be chosen with a certain physical understanding [9]. In a further step, only a small amount of calculation time is necessary to determine a small quantity of significant constants in order to bring the calculated values into line with the measured ones.…”