def prob3():
def bvp(epsilon, subintervals):
# for figure3.pdf
X, Y = fd_order2_ode(
func=lambda x: np.cos(x),
a1=lambda x: epsilon,
a2=lambda x: 0.0,
a3=lambda x: -4.0 * (np.pi - x ** 2.0),
a=0.0,
b=np.pi / 2.0,
alpha=0.0,
beta=1.0,
N=subintervals,
)
return X, Y
eps, subintervals = 0.1, 10
X, Y = bvp(eps, subintervals)
plt.plot(X, Y, "-k", mfc="None", linewidth=2.0)
plt.ylabel("$y$", fontsize=16)
plt.xlabel("$x$", fontsize=16)
# plt.axis([-.1,np.pi/2.+.1,-.1,1.5])
# plt.savefig('figure3.pdf')
plt.show()
plt.clf()
raise SystemError
num_approx = 6 # Number of Approximations
N = 2560 * np.array([2 ** j for j in range(num_approx)])
approx_order(num_approx, N, bvp, eps)
return
def prob2():
def bvp(epsilon, subintervals):
# for figure2.pdf
X, Y = fd_order2_ode(
func=lambda x: -1.0,
a1=lambda x: epsilon,
a2=lambda x: -1.0,
a3=lambda x: 0.0,
a=0.0,
b=1.0,
alpha=1.0,
beta=3.0,
N=subintervals,
)
return X, Y
def AnalyticSolution(x, alpha, beta, epsilon):
out = alpha + x + (beta - alpha - 1.0) * (np.exp(x / epsilon) - 1.0) / (np.exp(1.0 / epsilon) - 1.0)
return out
eps, subintervals = 0.1, 20
X, Y = bvp(eps, subintervals)
plt.plot(X, Y, "-k", mfc="None", linewidth=2.0)
plt.ylabel("$y$", fontsize=16)
plt.xlabel("$x$", fontsize=16)
# plt.axis([-.1,1.1,1-.1,3+.1])
# plt.savefig('figure2.pdf')
plt.show()
plt.clf()
num_approx = 6 # Number of Approximations
N = 2560 * np.array([2 ** j for j in range(num_approx)])
approx_order(num_approx, N, bvp, eps)
return
def prob3():
def bvp(epsilon, subintervals):
# for figure3.pdf
X, Y = fd_order2_ode(func=lambda x: np.cos(x),
a1=lambda x: epsilon,
a2=lambda x: 0.,
a3=lambda x: -4. * (np.pi - x**2.),
a=0.,
b=np.pi / 2.,
alpha=0.,
beta=1.,
N=subintervals)
return X, Y
eps, subintervals = 0.1, 10
X, Y = bvp(eps, subintervals)
plt.plot(X, Y, '-k', mfc="None", linewidth=2.0)
plt.ylabel('$y$', fontsize=16)
plt.xlabel('$x$', fontsize=16)
# plt.axis([-.1,np.pi/2.+.1,-.1,1.5])
# plt.savefig('figure3.pdf')
plt.show()
plt.clf()
raise SystemError
num_approx = 6 # Number of Approximations
N = 2560 * np.array([2**j for j in range(num_approx)])
approx_order(num_approx, N, bvp, eps)
return
def prob2():
def bvp(epsilon, subintervals):
# for figure2.pdf
X, Y = fd_order2_ode(func=lambda x: -1.,
a1=lambda x: epsilon,
a2=lambda x: -1.,
a3=lambda x: 0.,
a=0.,
b=1.,
alpha=1.,
beta=3.,
N=subintervals)
return X, Y
def AnalyticSolution(x, alpha, beta, epsilon):
out = alpha + x + (beta - alpha - 1.) * (np.exp(x / epsilon) - 1.) / (
np.exp(1. / epsilon) - 1.)
return out
eps, subintervals = 0.1, 20
X, Y = bvp(eps, subintervals)
plt.plot(X, Y, '-k', mfc="None", linewidth=2.0)
plt.ylabel('$y$', fontsize=16)
plt.xlabel('$x$', fontsize=16)
# plt.axis([-.1,1.1,1-.1,3+.1])
# plt.savefig('figure2.pdf')
plt.show()
plt.clf()
num_approx = 6 # Number of Approximations
N = 2560 * np.array([2**j for j in range(num_approx)])
approx_order(num_approx, N, bvp, eps)
return
def prob5():
def bvp(epsilon, subintervals):
# X,Y = fd_order2_ode(func=lambda x: 0.,a1=lambda x:1.,
# a2=lambda x: 4.*x/(epsilon+x**2.),a3=lambda x:2./(epsilon+x**2.),
# a=-1.,b=1., alpha=1./(1.+epsilon),
# beta=1./(1.+epsilon),N=subintervals)
X,Y = fd_order2_ode(func=lambda x: 0.,a1=lambda x:(epsilon+x**2.),
a2=lambda x: 4.*x,a3=lambda x:2.,
a=-1.,b=1., alpha=1./(1.+epsilon),
beta=1./(1.+epsilon),N=subintervals)
return X,Y
eps, subintervals = 0.01, 100
X,Y = bvp(eps, subintervals)
plt.plot(X,Y,'-k',mfc="None",linewidth=2.0)
# eps, subintervals = 0.02, 200
# X,Y = bvp(eps, subintervals)
# plt.plot(X,Y,'-k',mfc="None",linewidth=2.0)
plt.ylabel('$y$',fontsize=16)
plt.xlabel('$x$',fontsize=16)
# plt.savefig('figure5.pdf')
plt.show()
plt.clf()
# import sys; sys.exit()
num_approx = 5 # Number of Approximations
N = 2560*np.array([2**j for j in range(num_approx)])
approx_order(num_approx,N,bvp,eps)
return
def prob4():
def bvp(epsilon, subintervals):
def g(x):
out = -epsilon*pi**2.*cos(pi*x) - pi*x*sin(pi*x)
return out
X,Y = fd_order2_ode(func=g,a1=lambda x:epsilon,
a2=lambda x: x,a3=lambda x:0.,
a=-1.,b=1., alpha=-2.,beta=0.,N=subintervals)
return X,Y
eps, subintervals = 0.1, 20
X,Y = bvp(eps, subintervals)
plt.plot(X,Y,'-k',mfc="None",linewidth=2.0)
import sys; sys.exit()
eps, subintervals = 0.01, 400
X,Y = bvp(eps, subintervals)
plt.plot(X,Y,'-k',mfc="None",linewidth=2.0)
eps, subintervals = 0.001, 400
X,Y = bvp(eps, subintervals)
plt.plot(X,Y,'-k',mfc="None",linewidth=2.0)
plt.ylabel('$y$',fontsize=16)
plt.xlabel('$x$',fontsize=16)
# plt.savefig('figure4.pdf')
plt.show()
plt.clf()
num_approx = 6 # Number of Approximations
N = 2560*np.array([2**j for j in range(num_approx)])
approx_order(num_approx,N,bvp,eps)
return