p32: Solve linear BVP

\[ u_{xx} = \exp(4x) \]

\[ u(-1)=0, \qquad u(1)=1 \]

Similar to p13 but with non-homogeneous bc.

%config InlineBackend.figure_format='svg'
from chebPy import *
from numpy import dot,exp,zeros,sinh,cosh,max,linspace,polyval,polyfit,inf
from numpy.linalg import norm
from scipy.linalg import solve
from matplotlib.pyplot import title,plot,xlabel,ylabel,grid

N = 16
D,x = cheb(N)
D2 = dot(D,D)
D2 = D2[1:N,1:N]
f = exp(4.0*x[1:N])
u = solve(D2,f)
s = zeros(N+1)
s[1:N] = u
s = s + (x + 1.0)/2.0 # Correction for bc

xx = linspace(-1.0,1.0,200)
uu = polyval(polyfit(x,s,N),xx)    # interpolate grid data
exact = (exp(4.0*xx) - sinh(4.0)*xx - cosh(4.0))/16.0 + (xx + 1.0)/2.0
maxerr = norm(uu-exact,inf)

title('max err = %e' % maxerr)
plot(x,s,'o',xx,exact)
xlabel('x'); ylabel('u'); grid(True);