Approximate Solutions of Differential Equations
The subject of this chapter is the derivation of the equations that are to be solved for an approximate analytical solution of a perturbation problem involving a differential equation. There are several ways to obtain an approximate analytical solution. First, approximations can be made in the equation such that the remaining equation has an analytical solution. If the approximate solution is accurate enough, work stops. If not, something more must be done. Second, instead of discarding the small terms, they can be replaced by a small parameter, thereby creating a perturbation problem. The effect of the size of the parameter can be investigated by simulation. Third, if the original equation contains a small parameter and if the equation with the parameter equal to zero has an analytical solution, a perturbation problem results. Fourth, a perturbation problem also arises when a small change in the initial conditions is introduced.
KeywordsFinal Time Point Mass Taylor Series Expansion Approximate Analytical Solution Nominal Path
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