Differences between revisions 10 and 12 (spanning 2 versions)
 ⇤ ← Revision 10 as of 2007-06-24 23:50:21 → Size: 1715 Editor: DavidJoyner Comment: ← Revision 12 as of 2007-06-24 23:58:15 → ⇥ Size: 2118 Editor: DavidJoyner Comment: Deletions are marked like this. Additions are marked like this. Line 39: Line 39: $$x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} )$$ $$x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} )$$ Line 43: Line 41: $$\ {\mapsto}\ 1 - ( x - 1 )^2$$ $$x \ {\mapsto}\ 1 - ( x - 1 )^2$$ Line 56: Line 52: }}}However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented:{{{sage: f(x) = x^2+1 sage: g(x) = 1-(x-1)^3sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])sage: P.laplace(x,s)(s^3 - 6)*e^(-s)/s^4 - ((2*s^2 + 2*s + 2)*e^(-s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(-3*s)/s^4 + (-3*s - 1)*e^(-3*s)/s^2 + (5*s + 1)*e^(-5*s)/s^2 + (s^2 + 2)/s^3

# Differential Equations

## Laplace Transform (LT) methods

### Unit Step Functions

SAGE can define piecewise functions like

x \ {\mapsto}\ \sin ( \frac{\pi \cdot x}{2} )
on (0, 1),
x \ {\mapsto}\ 1 - ( x - 1 )^2
on (1, 3),
x \ {\mapsto}\ -x
on (3, 5), as follows:

sage: f(x) = sin(x*pi/2)
sage: g(x) = 1-(x-1)^2
sage: h(x) = -x
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: latex(P)

However, at the moment only Laplace transforms of "piecewise polynomial" functions are implemented:

sage: f(x) = x^2+1
sage: g(x) = 1-(x-1)^3
sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]])
sage: P.laplace(x,s)
(s^3 - 6)*e^(-s)/s^4 - ((2*s^2 + 2*s + 2)*e^(-s)/s^3) + (7*s^3 + 12*s^2 + 12*s + 6)*e^(-3*s)/s^4 + (-3*s - 1)*e^(-3*s)/s^2 + (5*s + 1)*e^(-5*s)/s^2 + (s^2 + 2)/s^3

## PDEs

### Heat Eqn. Both Ends Insulated

Differential_Equations (last edited 2008-11-14 13:42:08 by localhost)