- #1
yungman
- 5,718
- 241
I have two questions:
(1)As the tittle, if [itex]u(a,\theta,t)=0[/itex], is
[tex]\frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}[/tex]
and
[tex]\frac{\partial^2{u}}{\partial {t}^2}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}[/tex]
Just Poisson Equation
[tex]\nabla^2u=h(r,\theta,t)[/tex]
Where
[tex] h(r,\theta,t)=\frac{\partial{u}}{\partial {t}}\;\hbox { or }\;h(r,\theta,t)=\frac{\partial^2{u}}{\partial {t}^2}\;\hbox{ respectively.}[/tex](2)AND if [itex]u(a,\theta,t)=f(r,\theta,t)[/itex], then we have to use superposition of Poisson with zero boundary plus Dirichlet with [itex]u(a,\theta,t)=f(r,\theta,t)[/itex]?
That is
[tex] u(r,\theta,t)=u_1+u_2[/tex]
where
[tex]\nabla^2u_1=h(r,\theta,t)\;\hbox { with }\;u(a,\theta,t)=0[/tex]
and
[tex]\nabla^2u_2=0\;\hbox { with }\;u(a,\theta,t)=f(r,\theta,t)[/tex]
Thanks
(1)As the tittle, if [itex]u(a,\theta,t)=0[/itex], is
[tex]\frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}[/tex]
and
[tex]\frac{\partial^2{u}}{\partial {t}^2}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2}[/tex]
Just Poisson Equation
[tex]\nabla^2u=h(r,\theta,t)[/tex]
Where
[tex] h(r,\theta,t)=\frac{\partial{u}}{\partial {t}}\;\hbox { or }\;h(r,\theta,t)=\frac{\partial^2{u}}{\partial {t}^2}\;\hbox{ respectively.}[/tex](2)AND if [itex]u(a,\theta,t)=f(r,\theta,t)[/itex], then we have to use superposition of Poisson with zero boundary plus Dirichlet with [itex]u(a,\theta,t)=f(r,\theta,t)[/itex]?
That is
[tex] u(r,\theta,t)=u_1+u_2[/tex]
where
[tex]\nabla^2u_1=h(r,\theta,t)\;\hbox { with }\;u(a,\theta,t)=0[/tex]
and
[tex]\nabla^2u_2=0\;\hbox { with }\;u(a,\theta,t)=f(r,\theta,t)[/tex]
Thanks
Last edited: