def ODE_Solver(x, x_f, n, y):
h = (x_f - x) / (n - 1)
x1 = x2 = x
y1 = y2 = y
y1_points = []
y1_points.append(y1)
y2_points = []
y2_points.append(y2)
x_points = []
x_points.append(x1)
for i in range(n - 1):
y1, x1 = RK_Method(x1, h, y1)
y2, x2 = Euler_Forward(x2, h, y2)
y1_points.append(y1)
y2_points.append(y2)
x_points.append(x1)
print(x_points)
plt.plot(x_points, y1_points, label='RK4')
plt.plot(x_points, y2_points, label='Euler-Forward')
plt.legend()
pl.prutorsaveplot(plt, 'plot2d.pdf')
def System_of_ODE(t, t_f, h, y_i, z_i):
w = np.zeros((2, 1))
w[0] = y_i
w[1] = z_i
f = F(w, t)
y_points = []
y_points.append(y_i)
z_points = []
z_points.append(z_i)
t_points = []
t_points.append(t)
while t < t_f:
w, t = RK_Method(t, h, w, f)
y_points.append(w[0])
z_points.append(w[1])
t_points.append(t)
print('Plotting Y v/s T and Z v/s T')
plt.plot(t_points, y_points)
plt.plot(t_points, z_points)
pl.prutorsaveplot(plt, 'plot2d.pdf')
def Shooting_Method(x, x_f, h, a, b, v1, v2, e, i_max):
i = 0
err = 1
while err >= e:
x_i = x #starting_x
y = np.zeros((2,1))
y[0] = a
y[1] = v1
while (x < x_f):
f = F(y, x)
y, x = RK_Method(x, h, y, f)
u11 = y[0] #value_at_boundary_using_v1
x = x_i
y = np.zeros((2,1))
y[0] = a
y[1] = v2
y_points = []
y_points.append(a)
x_points = []
x_points.append(x)
while (x < L):
f = F(y, x)
y, x = RK_Method(x, h, y, f)
x_points.append(x)
y_points.append(y[0])
u12 = y[0] #value_at_boundary_using_v2
v3 = Secant_Method(v1, v2, b, u11, u12)
v1, v2 = v2, v3
err = abs(u12-b)
i = i+1
x = x_i
if i > i_max :
print('ERROR: Limit Exceeded')
return None
print('The solution was obtained after', i, 'iterations')
plt.plot(x_points, y_points)
pl.prutorsaveplot(plt, 'plot2d.pdf')
return None
def Direct_Method(a, b, h, l, flag):
n = int(l / h)
x = np.arange(n + 1) * h
L = np.zeros(n)
D = np.zeros(n)
U = np.zeros(n)
B = np.zeros(n)
for i in range(0, n):
L[i] = p(x[i + 1]) / (h**2) - (q(x[i + 1]) / (2 * h))
for i in range(0, n):
D[i] = -2 * p(x[i + 1]) / (h**2) + r(x[i + 1])
for i in range(0, n):
U[i] = p(x[i + 1]) / (h**2) + (q(x[i + 1]) / (2 * h))
for i in range(0, n):
B[i] = s(x[i + 1])
B[0] = B[0] - L[0] * a
if flag == 1:
print('Backward Difference')
L[n - 2] = L[n - 2] - (U[n - 2] / 3)
D[n - 2] = D[n - 2] + (4 * U[n - 2] / 3)
B[n - 2] = B[n - 2] - (2 * U[n - 2] * b * h / 3)
L[0] = U[n - 1] = L[n - 1] = D[n - 1] = B[n - 1] = 0
Y = np.array(a)
Y = np.append(Y, Thomas_Algo(L, D, U, B, flag))
Yn = (2 * b * h / 3) + (4 * Y[-1] / 3) - (Y[-2] / 3)
Y = np.append(Y, Yn)
print(Y)
print(x)
if flag == 0:
print('Ghost Node')
L[n - 1] = L[n - 1] + U[n - 1]
B[n - 1] = B[n - 1] - (2 * U[n - 1] * b * h)
L[0] = U[n - 1] = 0
Y = Thomas_Algo(L, D, U, B, flag)
Y = np.array(a)
Y = np.append(Y, Thomas_Algo(L, D, U, B, flag))
print(Y)
print(x)
plt.plot(X, Y)
plt.scatter(X, Y, c='k')
pl.prutorsaveplot(plt, 'plot2d.pdf')
def Plot(Q, x, y, a, Value): #Plot the Spline
n = len(x)
for i in range(n-1):
X = np.linspace(x[i], x[i+1], 100)
Y = Cubic_Spline_Value(Q, x, i, X)
plt.plot(X, Y)
plt.tight_layout()
plt.scatter(x, y, s=10, color='k', zorder=10)
if Value:
plt.scatter(a, Value, s=10, color='blue', zorder=10) #Plot the point in blue if its inside data range
pl.prutorsaveplot(plt, 'plot2d.pdf')
def Plots(x, u, ue, t, errmax, phi):
rcParams['figure.figsize'] = 5, 3
plt.figure()
plt.plot(x,u, '+r',label = r'FTCS solution')
plt.plot(x,ue, label = r'Analytic solution')
plt.xlabel(r'distance', fontsize=10)
plt.ylabel(r'function', fontsize=10)
pl.prutorsaveplot(plt, 'plot1.pdf')
plt.figure()
plt.plot(t,errmax)
plt.xlabel(r'time', fontsize=10)
plt.ylabel(r'maxerror', fontsize=10)
pl.prutorsaveplot(plt, 'plot2.pdf')
dist, time = np.meshgrid(x, t) # no idea why to do that
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(dist, time, np.transpose(phi),cmap=cm.jet, linewidth=0.1)
pl.prutorsaveplot(plt, 'plot3.pdf')
# Function to give spline values at different x values
def Qvalue(r, A, B, C, D):
q_i = 0
for i in range(n):
if (r > X[i]):
q_i = i
q = A[q_i] * ((r - X[q_i])**3) - B[q_i] * ((
r - X[q_i + 1])**3) + C[q_i] * (r - X[q_i]) - D[q_i] * (r - X[q_i + 1])
return q
# code for prutor plot
X_spline = np.linspace(X[0], X[n - 1], 100)
Y_spline = np.zeros(shape=(len(X_spline)))
for i in range(len(X_spline)):
Y_spline[i] = Qvalue(X_spline[i], A, B, C, D)
plt.plot(X,
Y,
color='green',
marker='o',
label='Original Data',
linestyle='None')
plt.plot(X_spline, Y_spline)
plt.legend(["Original Data", "Cubic Spline"])
plt.grid()
plt.xlabel('x')
plt.ylabel('F(x)')
pl.prutorsaveplot(plt, 'CubicSpline.pdf')