Differential Equations with Switching Conditions

The conditions under which a discontinuity occurs are in general known and can be formulated as roots of a vector-valued, time and state dependent switching function 

In Example 6.1.1 the switching functions are simple time functions Example 6.2.1 The problem 

For the numerical treatment of ODEs with discontinuities it is necessary to rewrite the differential equation as 

Assumption 6.2.2 For fixed s, f is assumed to be smooth with I being sufficiently large. 

Discontinuities can therefore occur only if a component of q changes its sign. 

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We will discuss differential equations with switching conditions and then present techniques for localizing roots of these functions. 

We are now ready to discuss the treatment of differential equations with switching conditions. The main task is the localization of the switching points as roots of the switching functions. Note, the complexity of this task depends on the form of the switching function. Switching functions may in general have the same complexity as the right hand side function. For example, an impact requires the computation of relative distances for both functions. 

After the perturbation is switched on, the wave function is no longer stationary and begins to evolve in time according to the time-dependent Schrodinger equation +V)ip = ih. This is a differential equation with partial derivatives with the boundary condition (x, t = 0) = The functions form... 

Equation (2.10) is a set of first-order differential equations which describe the evolution of the molecular system under the external perturbation h(t). It is valid for time-independent as well as time-dependent perturbations and must be solved subject to the initial conditions a (0) = 1 and oa j(0) = 0 if the molecule is initially in state F ). The time dependence of the coefficients aa(t) together with the stationary basis functions Fa(Q,q) describe completely the state of the molecule at each instant t. When the perturbation is switched off, the coefficients aa(t) become constant again. 

This is a simple differential equation that requires only two initial conditions as required by the physical problem. From a mathematical point of view, it is possible to switch from the first to the second formulation using x = L sin (cp) and y = -L cos (q ), but the first formulation will not be entertained by anyone familiar with the physical problem for several reasons. 

This case represents the least onerous duty for the switchgear. The angle (j> becomes small as the resistance increases. The worst-case switching angle 9 approaches zero. The conditions that produce a minimum or a maximum can be found by differentiating i in equation (11.5) with respect to the time t and equating the result to zero. This yields the following conditions,... 

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