Hosepipe.m

%% Combining Chebyshev and trigonometric
% Nick Trefehen, November 2019

%%
% (Chebfun example approx2/Hosepipe.m)

%% 1. Hosepipe
% Here is a surface you might find on your vacuum
% cleaner or under the hood of your car:
r = chebfun(@(x) .5+.04*cos(40*x));
F = chebfun2(@(x,phi) 2*x, 'trigy');
G = chebfun2(@(x,phi) r(x).*cos(pi*phi), 'trigy');
H = chebfun2(@(x,phi) r(x).*sin(pi*phi), 'trigy');
surf(F,G,H), axis equal off, camlight

%%
% Viewed as a surface, we see that this is periodic in the $\phi$
% direction and nonperiodic in the $x$ direction.  What's interesting
% is that its representation as a trio of chebfun2 objects shares
% this property: each of them is nonperiodic in the first variable
% and periodic in the second, because the flag |'trigy'| has
% been specified.  

%% 
% Until recently, a chebfun2 representation had to be Chebyshev
% in both variables or, if |'trig'| was specified,
% trigonometric in both directions.  The ability to mix the
% two with |'trigx'| or |'trigy'| is new.  Here we see
% some details of the three chebfun2 objects:
F, G, H

%%
% The |F| chebfun2 is trivial, but
% |G| is interesting.  The command |plotcoeffs| shows how it
% mixes different representations in the two directions.
% The |H| chebfun2 looks much the same.
plotcoeffs(G)

%%
% One reason for representing a periodic function periodically
% is that it is somewhat more efficient.  More important
% in practice is that its derivatives may retain their
% smoothness and accuracy at the wraparound point.

%% 2. Annulus
% The hosepipe example is relatively complicated.  A simpler illustration
% of the Cheb/trig combination is the discretization of a function 
% defined in an annulus: periodic in the angular direction, nonperiodic
% in the radial direction.  For example, the analytic function
% $f(z) = (1+4/z^3)^{-1} (z^3+0.1)^{-1}$ has three poles outside the unit
% disk and three inside.  For $1/2 \le |z| \le 3/2$ it is analytic,
% however, so its abolute value $F(r,t) = |f(re^{i t})|$ is smooth:
f = @(z) (1+4./z.^3).^-1.*(z.^3+.1).^-1;
F = @(r,t) abs(f(r.*exp(1i*t)));

%%
% Here we make a cheb/trig chebfun2 of $F$:
Fc = chebfun2(@(r,t) F(r,t),[.5 1.5 -pi pi],'trigy');

%%
clf, plot(Fc), colorbar
xlabel r, ylabel t

%%
% (I am not sure if Chebfun offers an elegant way to plot this
% in an annulus.)  Here are the Chebyshev and Fourier expansion
% coefficients:
plotcoeffs(Fc)