BinousShaikhBellagi.m

%% Problems from Binous, Shaikh and Bellagi
% Nick Trefethen, September 2014

%%
% (chebfun example temp/BinousShaikhBellagi)
% [Tags: #chebop]

%%
% An interesting new paper by Binous, Shaikh, and Bellagi explores
% various problems that can be solved by Chebyshev spectral methods [1].
% Here we solve a number of these problems with Chebfun, but not all
% of them; perhaps others could extend the discussion?
% The equation numbers are from [1], and we abbreviate the
% authors by ``BSB''.

%% 1. Split boundary value problem
% The first problem BSB address is (3),
%
% $$ uu'-u'' = 1, \quad u(-1) = 0,~ u(1) = 2 . $$
%
% BSB solve this by setting up a nonlinear system of equations
% based on Chebyshev collocation, then solving the system
% by FindRoot in Mathematica or fsolve in MATLAB.
% We can solve it in Chebfun by
N = chebop(-1,1);
N.op = @(x,u) u.*diff(u) - diff(u,2);
N.lbc = 0; N.rbc = 2;
u = N\1;

%%
% A plot agrees nicely with Figure 1 of [1],
plot(u), ylim([0 2]), grid on

%%
% and the values at $x=0$ and $x=1/\sqrt 2$
% agree with BSB Table 1,
format long
u([0 1/sqrt(2)])

%% 2. Diffusion problem
% Next BSB consider the PDE (5)
%
% $$ u_t = u_{yy} , \quad u(t,0)=u(t,1) = 0, ~u(0,x) = 1 , $$
%
% and they report the solution at time $t=0.0126$ based
% on NDSolve in Mathematica or ode15s in MATLAB.  We can do
% this as follows (type help chebop/expm for details):
L = chebop(0,1);
L.op = @(y,u) diff(u,2);
L.bc = 'dirichlet';
u0 = chebfun('1',[0 1]);
t = 0.0126;
u1 = expm(L,t,u0);

%%
% A plot agrees nicely with BSB Figure 2,
plot(u1), grid on

%%
% and the value at $y=0.5$ matches their Table 2: 
u1(0.5)

%% 3. Arnold problem
% The third problem addressed by BSB is (8)
%
% $$ c_t = Ac_{zz} + B c_{zz} , \quad c(t,0)=C,
% ~~c(t,\infty) = 0, ~~c(0,z) = 0 , $$
%
% where $A$, $B$ and $C$ are constants,
% describing evaporation of a liquid.  It seems
% that BSB don't list the values of the constants,
% however, so I will leave this one for now.

%% 4. Unsteady convection-diffusion problem
% The next problem is (11)
%
% $$ c_t = 0.49 Dc_{xx} - 2.5  c_x , \quad c(t,0)=1,
% ~~c(t,\infty) = 0, ~~c(0,z) = 0 , $$
%
% with $D = 0.49$.  The following formulation doesn't
% work; more thought needed.
dom = [0,10];
L = chebop(dom);
L.op = @(x,c) 0.49*diff(c,2) - 2.5*diff(c);
L.lbc = 1; L.rbc = 0;
c0 = chebfun('0',dom);
t = 1.33;    
u1 = expm(L,t,c0);
plot(u1)

%% 5. Falkner-Skan equation
% Next we look at
%
% $$ f''' + ff'' + (\pi/4)(1-(f')^2) = 0, \quad f(0) = f'(0) = 0, ~~
% f'(\infty) = 1 . $$
%
% BSB solve this on $[0,4]$ and then plot the derivative,

%%
dom = [0 4];
N = chebop(dom);
N.op = @(x,f) diff(f,3) + f.*diff(f,2) + (pi/4)*(1-diff(f).^2);
N.lbc = @(f) [f; diff(f)];
N.rbc = @(f) diff(f)-1;
f = N\0;
plot(diff(f)), grid on

%% 6. Non-Newtonian Carreau fluid
% BSB now consider a non-Newtonian fluids problem with
% a shear-dependent viscosity mu,
%
% $$ mu(g) = 0.00204 (1+ 0.04 g^2)^{-1/4} $$
%
% where $g = \dot\gamma$ is the shear rate.
% I am not sure what they do with this formula.

%% 7. Convection past an isothermal plate
% This problem involves a couples system of two ODEs
% involving the stream
% function $F(y)$ and the temperature $T(y)$:
% 
% $$ F''' + 3FF' - 2(F')^2 + T = 0, \quad 
% T'' + 30FT' = 0, $$
%
% with five boundary conditions,
%
% $$ F(0) = F'(0) = 0, ~~ F'(\infty) = 0, ~~
% T(0) = 1, ~~ T(\infty) = 0. $$
% 
% They approximate $[0,\infty]$ by $[0,5]$, but my attempt with
% this interval fails, so I will shorten it to $[0,2.5]$:
dom = [0 2.5];
N = chebop(dom);
N.op = @(y,F,T) [diff(F,3)+3*F.*diff(F)-2*diff(F).^2+T; diff(T,2)+30*F.*diff(T)];
N.lbc = @(F,T) [F; diff(F); T-1];
N.rbc = @(F,T) [diff(F); T];
U = N\0;

%%
% BSB plot the temperature,
T = U{2}; plot(T), grid on, title temperature
%%
% and the velocity, which is $F'$,
F = U{1}; plot(diff(F)), grid on, title velocity

%% Reference
%
% 1. H. Binous, A. A. Shaikh, and A. Bellagi, ``Chebyshev orthogonal
%    collocation technique to solve transport phenomena problems with Matlab
%    and Mathematica", _Computer Applications in Engineering Education_, 2014,
%    pp. 1-10.