

In [11]:

    
using Plots,ApproxFun
plotlyjs()  # works in GR too;

Poisson equation $u_{\theta\theta} + u_{\phi\phi} = f(\theta,\phi)$



In [12]:

    
d=PeriodicInterval()^2
f=Fun((θ,ϕ)->exp(-10(sin(θ/2)^2+sin(ϕ/2)^2)),d)
A=lap(d)+.1I
u=A\f
plot(u)









    Out[12]:

Laplace equation on the periodic strip $u_{\theta\theta} + u_{yy} = 0$



In [13]:

    
d=PeriodicInterval()*Interval()
g=Fun(z->real(cos(z)),∂(d))  # boundary data
u=[dirichlet(d);Laplacian(d)]\g
plot(u)









    Out[13]:

Transport equation $u_t + u_\theta = 0$, $\theta \in (-2,2)$



In [15]:

    
dθ=PeriodicInterval(-2.,2.);dt=Interval(0,3.)
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])
u=[I⊗ldirichlet(dt);Dt+Dθ]\Fun(θ->exp(-20θ^2),dθ)
plot(u)









    Out[15]:

$u_t + (1+\cos \theta) u_\theta = 0$



In [16]:

    
dθ=PeriodicInterval();dt=Interval(0,2.)
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])
c=1+Fun(cos,dθ)

#timedirichlet is [u[x,0], u[-1,t], u[1,t]

u=[I⊗ldirichlet(dt);Dt+c*Dθ]\Fun(θ->exp(-20θ^2),dθ)
plot(u)









    Out[16]:

$u_t = \sin \theta \,u_\theta$



In [6]:

    
dθ=PeriodicInterval();dt=Interval(0,2.)
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])
a=Fun(sin,dθ)

#timedirichlet is [u[x,0], u[-1,t], u[1,t]

u=[I⊗ldirichlet(dt);Dt-a*Dθ]\Fun(θ->exp(-20θ^2),dθ)
plot(u)









    Out[6]:

Convection Diffusion $u_t = \epsilon u_{\theta\theta} + u_\theta$



In [7]:

    
dθ=PeriodicInterval();dt=Interval(0,10.)
d=dθ*dt
ε=.01
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])

# Parentheses are a hack to get rank 2 PDE
u=[I⊗ldirichlet(dt);Dt-ε*Dθ^2-Dθ]\Fun(θ->exp(-20θ^2),dθ)
plot(u)









    Out[7]:

Wave equation $u_{tt} = u_{\theta\theta}$



In [8]:

    
dθ=PeriodicInterval(-5.,5.);dt=Interval(0,20.)
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])
# need to specify both ic and its derivative
B=[I⊗ldirichlet(dt);I⊗lneumann(dt)]
u=pdesolve([B;Dt^2-Dθ^2],Fun(θ->exp(-20(θ-.1)^2),dθ),200)
plot(u)









    Out[8]:

Linear KdV



In [9]:

    
dθ=PeriodicInterval(-10.0,2.);dt=Interval(0,.03);
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1])
u=pdesolve([I⊗ldirichlet(dt);Dt+Dθ^3],Fun(θ->exp(-10θ^2),dθ),200)
plot(u)









    Out[9]:

Beam equation $$u_{tt}+u_{\theta\theta\theta\theta}=0$$



In [10]:

    
dθ=PeriodicInterval(0.0,1.0);dt=Interval(0,0.03)
d=dθ*dt
Dθ=Derivative(d,[1,0]);Dt=Derivative(d,[0,1]);

B=[I⊗ldirichlet(dt);I⊗lneumann(dt)]
u=pdesolve([B;Dt^2+Dθ^4],Fun(θ->exp(-200(θ-.5).^2),dθ),200)
plot(u)









    Out[10]: