How to solve 2nd order BVP using Chebyshev Spectral Method?

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Muhammad Usman 2023-3-27

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https://ww2.mathworks.cn/matlabcentral/answers/1936069-how-to-solve-2nd-order-bvp-using-chebyshev-spectral-method

编辑： Torsten 2023-3-28

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Hi, I am tring to solve the following BVP using Chebshev Spectral Method:

with boundary conditions:

I am following the code written by Lloyd N. Trefethen in his book "SPECTRAL METHODS IN MATLAB".

The code is given by for Linear BVP

% p33.m - solve linear BVP u_xx = exp(4x), u'(-1)=u(1)=0
N = 16;
[D,x] = cheb(N); D2 = D^2;              
D2(N+1,:) = D(N+1,:);            % Neumann condition at x = -1
D2 = D2(2:N+1,2:N+1);                
f = exp(4*x(2:N));           
u = D2\[f;0];                       
u = [0;u];              
clf, subplot('position',[.1 .4 .8 .5])
plot(x,u,'.','markersize',16)
axis([-1 1 -4 0])
xx = -1:.01:1;
uu = polyval(polyfit(x,u,N),xx);   
line(xx,uu)
grid on
exact = (exp(4*xx) - 4*exp(-4)*(xx-1) - exp(4))/16; 
title(['max err = ' num2str(norm(uu-exact,inf))],'fontsize',12)

where cheb was defined as:

% CHEB  compute D = differentiation matrix, x = Chebyshev grid
function [D,x] = cheb(N)
if N==0, D=0; x=1; return, end
x = cos(pi*(0:N)/N)'; 
c = [2; ones(N-1,1); 2].*(-1).^(0:N)';
X = repmat(x,1,N+1);
dX = X-X';                  
D  = (c*(1./c)')./(dX+(eye(N+1)));      % off-diagonal entries
D  = D - diag(sum(D'));                 % diagonal entries