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In [1]:



    


# Requires Gadfly
using ApproxFun

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





In [3]:



    


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
ApproxFun.contour(u)



    


















    
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    y

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





In [4]:



    


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



    


















    
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    y

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





In [5]:



    


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



    


















    
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    y

We can interchange the variables using .'





In [6]:



    


A=[ldirichlet(dt)⊗I;(Dt+Dθ).']
f=Fun(θ->exp(-20θ^2),dθ)
u=A\f
ApproxFun.contour(u)



    


















    
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    -3.0
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    0.0
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    1.0
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    1.8
    2.0
    2.2
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    2.8
    3.0
    3.2
    3.4
    3.6
    3.8
    4.0
    4.2
    4.4
    4.6
    4.8
    5.0
    5.2
    5.4
    5.6
    5.8
    6.0
  
  
    
      0.8
      1.0
      0.4
      0.6
      0.0
      0.2
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
        
        
        
        
        
        
      
    
    
      Color
    
  
  
    
      
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
    
      
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
      
    
    
      
        
        
      
      
        
      
      
        
      
      
        
        
      
    
  
  
    -7
    -6
    -5
    -4
    -3
    -2
    -1
    0
    1
    2
    3
    4
    5
    6
    7
    -6.0
    -5.8
    -5.6
    -5.4
    -5.2
    -5.0
    -4.8
    -4.6
    -4.4
    -4.2
    -4.0
    -3.8
    -3.6
    -3.4
    -3.2
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
    3.2
    3.4
    3.6
    3.8
    4.0
    4.2
    4.4
    4.6
    4.8
    5.0
    5.2
    5.4
    5.6
    5.8
    6.0
    -6
    -3
    0
    3
    6
    -6.0
    -5.5
    -5.0
    -4.5
    -4.0
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    6.0
  
  
    y

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





In [7]:



    


dθ=PeriodicInterval();dt=Interval(0,2.)
d=dθ*dt
Dθ=Derivative(d,1);Dt=Derivative(d,2)
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)
ApproxFun.contour(u)



    


















    
Out[7]:









  
    x
  
  
    -14
    -12
    -10
    -8
    -6
    -4
    -2
    0
    2
    4
    6
    8
    10
    12
    14
    -12.0
    -11.5
    -11.0
    -10.5
    -10.0
    -9.5
    -9.0
    -8.5
    -8.0
    -7.5
    -7.0
    -6.5
    -6.0
    -5.5
    -5.0
    -4.5
    -4.0
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    6.0
    6.5
    7.0
    7.5
    8.0
    8.5
    9.0
    9.5
    10.0
    10.5
    11.0
    11.5
    12.0
    -20
    -10
    0
    10
    20
    -12.0
    -11.5
    -11.0
    -10.5
    -10.0
    -9.5
    -9.0
    -8.5
    -8.0
    -7.5
    -7.0
    -6.5
    -6.0
    -5.5
    -5.0
    -4.5
    -4.0
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    6.0
    6.5
    7.0
    7.5
    8.0
    8.5
    9.0
    9.5
    10.0
    10.5
    11.0
    11.5
    12.0
  
  
    
      0.25
      0.75
      0.50
      0.00
      1.00
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
        
        
        
        
        
      
    
    
      Color
    
  
  
    
      
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
    
    
      
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
      
    
    
      
        
        
      
      
        
      
      
        
      
      
        
        
      
    
  
  
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    -2.0
    -1.9
    -1.8
    -1.7
    -1.6
    -1.5
    -1.4
    -1.3
    -1.2
    -1.1
    -1.0
    -0.9
    -0.8
    -0.7
    -0.6
    -0.5
    -0.4
    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    3.1
    3.2
    3.3
    3.4
    3.5
    3.6
    3.7
    3.8
    3.9
    4.0
    -2
    0
    2
    4
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
    3.2
    3.4
    3.6
    3.8
    4.0
  
  
    y

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





In [8]:



    


dθ=PeriodicInterval();dt=Interval(0,2.)
d=dθ*dt
Dθ=Derivative(d,1);Dt=Derivative(d,2)
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)
ApproxFun.contour(u)



    


















    
Out[8]:









  
    x
  
  
    -14
    -12
    -10
    -8
    -6
    -4
    -2
    0
    2
    4
    6
    8
    10
    12
    14
    -12.0
    -11.5
    -11.0
    -10.5
    -10.0
    -9.5
    -9.0
    -8.5
    -8.0
    -7.5
    -7.0
    -6.5
    -6.0
    -5.5
    -5.0
    -4.5
    -4.0
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    6.0
    6.5
    7.0
    7.5
    8.0
    8.5
    9.0
    9.5
    10.0
    10.5
    11.0
    11.5
    12.0
    -20
    -10
    0
    10
    20
    -12.0
    -11.5
    -11.0
    -10.5
    -10.0
    -9.5
    -9.0
    -8.5
    -8.0
    -7.5
    -7.0
    -6.5
    -6.0
    -5.5
    -5.0
    -4.5
    -4.0
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    4.0
    4.5
    5.0
    5.5
    6.0
    6.5
    7.0
    7.5
    8.0
    8.5
    9.0
    9.5
    10.0
    10.5
    11.0
    11.5
    12.0
  
  
    
      0.25
      0.75
      0.50
      0.00
      1.00