Cadabra manual: Ndsolve

Damped harmonic oscillator

This is the harmonic oscillator with friction example of the Boost ODEint manual.

from cdb.graphics.plot import *

\tau::Coordinate; \dot{#}::Derivative(\tau); {x,y}::Depends(\tau);

$\displaystyle{}\text{Property Coordinate attached to }\tau.$

$\displaystyle{}\text{Property Derivative attached to }\backslash\texttt{dot}\{\#\}.$

$\displaystyle{}\text{Property Depends attached to }\left[x, y\right].$

ho:= [ \dot{x} = y, \dot{y} = -x + \gamma y ];

$\displaystyle{}\left[\dot{x} = y, \dot{y} = -\,x +\gamma y\right]$

substitute(ho, $\gamma->-0.2$);

$\displaystyle{}\left[\dot{x} = y, \dot{y} = -\,x -0.2\,y\right]$

p=None out=None

for gamma in np.linspace(0, -0.3, 50): sol = ndsolve($[ \dot{x} = y, \dot{y} = -x + @(gamma) y ]$, {$x$:0, $y$:1}, ($\tau$, 0, 10)) p = plot(sol, {$\tau$: (0,10)}, fig = p) out = display(p, out)

Lotka-Volterra equations

The following example is a little bit more complicated. It shows how to numerically solve a system of two ODEs, and then adds a slider so you can interact with the value of the constant which appears in the equations.

from cdb.interact.slider import * from cdb.graphics.plot import * fig=None

t::Coordinate. {x, y}::Depends(t). \dot{#}::Derivative(t). K = 0.45

The equations themselves describe a very simple predator-prey system, in which $x$ are the number of prey and $y$ are the number of predators.

lotka_volterra := [ \dot{x} = x*(1-y), \dot{y} = - @(K) y*(1-x) ];

$\displaystyle{}\left[\dot{x} = x \left(1 -\,y\right), \dot{y} = -0.73\,y \left(1 -\,x\right)\right]$

We solve these numerically with some initial conditions, and then feed the solutions of ndsolve straight into plot to visualise them.

solution = ndsolve( lotka_volterra, {$x$: 0.6, $y$: 0.2}, ($t$, 0, 30) ) fig = plot(solution, {$t$: (0,30)}, title=f"Lotka-Volterra with K={K}", fig=fig, yrange=[0, 5]);

slider("K", K, 0.1, 1.5, 0.05);

If you drag the slider above, the numerical solution to the ODEs will be re-computed and then re-plotted.