Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

In Von Neumann's stability analysis of Finite Difference Equations, the Euler's formula is used to describe a perturbation. Now the finite difference equation is stable if this perturbation does not grow in time.

Now, my question really is: how do I plot the perturbation (Equation 1) three-dimensions(?)/Im vs Real dimensions?

$$\epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x} \ldots \ldots \text{(1)}$$

I realize that the the Euler's formula $e^{ik_m x}$ describes a circle. Clearly the multiplicative amplitude factor $e^{at}$ either "grows" this circle or decreases it whether or not $e^{at}$ is greater than 1 or not.

I would like to describe that as a Manipulate or a ListAnimate in Mathematica but I don't understand which function to use.

So far I have understood that PolarPlot and ParametricPlot draw this circle in 2D.

Parametric Plot example

ParametricPlot[G {Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, 2 \[Pi]}, 
 PlotRange -> {{-1, 1}, {-1, 1}}]

enter image description here

Polar Plot example

PolarPlot[{Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, Pi}]

enter image description here

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

Your question asks, in part, "Clearly the multiplicative amplitude factor $e^{at}$ either "grows" this circle or decreases it whether or not $e^{at}$ is greater than 1 or not. I would like to describe that as a Manipulate or a ListAnimate in Mathematica." Here's one way to visualize the growing and/or shrinking.

Manipulate[
 ParametricPlot[a {Re[Exp[I x]], Im[Exp[I x]]}, {x, 0, 2 \[Pi]}, 
   PlotRange -> {{-1, 1}, {-1, 1}}], {a, 0.1, 2}]

This draws the circle and let's you change the radius by moving the slider to specify different a values.

enter image description here

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.