

In [1]:

    
# Requires PyPlot

using ApproxFun

Laplace Equation $u_{xx} + u_{yy} = 0$



In [2]:

    
d=Disk()
u=[dirichlet(d),lap(d)]\Fun(z->real(exp(z)),Circle())

ApproxFun.plot(pad(u,50,50)) # we pad to get a nice plot;









    



INFO: Loading help data...

Check the error:



In [3]:

    
u[.1,.2]-real(exp(.1+.2im))









    Out[3]:





0.0 + 1.3877787807814457e-17im

Poisson equation $u_{xx} + u_{yy} = f(x,y)$



In [9]:

    
d=Disk()
f=Fun((x,y)->exp(-10(x+.2)^2-20(y-.1)^2),d) 
u=[dirichlet(d),lap(d)]\[0.,f]
ApproxFun.plot(u);

Helmholtz $u_{xx} + u_{yy} + 100u=0$



In [5]:

    
d=Disk()

u=[dirichlet(d),lap(d)+100I]\1.0
ApproxFun.plot(u);



In [6]:

    
d=Disk()

u=[neumann(d),lap(d)+100I]\1.0
ApproxFun.plot(u);

Screened Poisson $u_{xx} + u_{yy} - 100u = 0, \partial u(\partial d) = 1$



In [7]:

    
d=Disk()
u=[neumann(d),lap(d)-100.0I]\1.0
ApproxFun.plot(u);

Laplacian squared $\Delta^2 u = 0$



In [8]:

    
d=Disk()

u=[dirichlet(d),neumann(d),lap(d)^2]\Fun(z->real(exp(z)),Circle())
ApproxFun.plot(pad(u,50,50)) # we pad to get a nice plot;