In sage, I performed the laplace transformation:
de=diff(y,t,2)-diff(y,x,2)==k*sin(pi*x);de.laplace(t,s)
and got such a code:
s^2L(y(x,t),t,s)-D[0](y)(x,t)2D[0,0]L(y(x,t),t,s)?sy(x,0)?D[0]L(y(x,t),t,s)D[0,0](y)(x,t)?D[1](y)(x,0)=sksin(?x)
Can someone tells me what D[0](y)(x,t)2D[0,0]L(y(x,t),t,s) and D[0]L(y(x,t),t,s)D[0,0](y)(x,t) mean?
Thanks!
Here is what I get.
var('y t s k')
y=function('y',x,t)
de=diff(y,t,2)-diff(y,x,2)==k*sin(pi*x)
de.laplace(t,s)
gives
s^2*laplace(y(x, t), t, s) - D[0](y)(x, t)^2*D[0, 0](laplace)(y(x, t), t, s) - s*y(x, 0) - D[0](laplace)(y(x, t), t, s)*D[0, 0](y)(x, t) - D[1](y)(x, 0) == k*sin(pi*x)/s
Within this expression, the meanings are:
$\frac{\partial y}{\partial t}$ = D[1](y)(x, t)
$\frac{\partial y}{\partial x}$ = D[0](y)(x, t)
$\frac{\partial^2 y}{\partial t^2}$ = D[1, 1](y)(x, t)
$\frac{\partial^2 y}{\partial x^2}$ = D[0, 0](y)(x, t)
So, D[0](y)(0, x) is $\frac{\partial y}{\partial t}|_{t=0}$.
You can check the first one above by doing diff(y,t).
What you're seeing is the displayed notation for symbolic partial derivatives. D[0] means "apply the derivative operator with respect to variable number 0" and D[0,0] represents two successive derivatives with respect to the variable. Observe:
sage: f = function('f', x)
sage: f.diff(x)
D[0](f)(x)
sage: f.diff(x).diff(x)
D[0, 0](f)(x)
Asked: 2012-08-16 22:29:28 +0100
Seen: 637 times
Last updated: Aug 19 '12
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