def initCond(u): # Init. values of [y,y']; use 'u' if unknown
return np.array([0.0, u])
def r(u): # Boundary condition residual--see Eq. (8.3)
X, Y = integrate(F, xStart, initCond(u), xStop, h)
y = Y[len(Y) - 1]
r = y[0] - 1.0
return r
def F(x, y): # First-order differential equations
F = np.zeros(2)
F[0] = y[1]
F[1] = -3.0 * y[0] * y[1]
return F
xStart = 0.0 # Start of integration
xStop = 2.0 # End of integration
u1 = 1.0 # 1st trial value of unknown init. cond.
u2 = 2.0 # 2nd trial value of unknown init. cond.
h = 0.1 # Step size
freq = 2 # Printout frequency
u = ridder(r, u1, u2) # Compute the correct initial condition
X, Y = integrate(F, xStart, initCond(u), xStop, h)
printSoln(X, Y, freq)
input("\nPress return to exit")
def initCond(u): # Initial values of [y,y',y",y"'];
# use 'u' if unknown
return array([0.0, 0.0, u[0], u[1]])
def r(u): # Boundary condition residuals-- see Eq. (8.7)
r = zeros(len(u),type=Float64)
X,Y = integrate(F,x,initCond(u),xStop,h)
y = Y[len(Y) - 1]
r[0] = y[2]
r[1] = y[3] - 1.0
return r
def F(x,y): # First-order differential equations
F = zeros((4),type=Float64)
F[0] = y[1]
F[1] = y[2]
F[2] = y[3]
if x == 0.0: F[3] = -12.0*y[1]*y[0]**2
else: F[3] = -4.0*(y[0]**3)/x
return F
x = 0.0 # Start of integration
xStop = 1.0 # End of integration
u = array([-1.0, 1.0]) # Initial guess for u
h = 0.1 # Initial step size
freq = 1 # Printout frequency
u = newtonRaphson2(r,u,1.0e-5)
X,Y = integrate(F,x,initCond(u),xStop,h)
printSoln(X,Y,freq)
raw_input("\nPress return to exit")
from printSoln import *
#F is array containing primes of variables x' and z', y is array of variables x,z
#dx/dt=z
def F(t,y):
F=numpy.zeros(2)
F[0]=y[1]
F[1]=(-w**2)*(y[0]**3)#this is saying z'=-w^2x^3, setting up differential equation.
return F
y=numpy.array([1,0])#initial conditions
w=1
t=0.#initial value of t
tStop=20.#final value of t
h=0.001#h=size of steps
freq=1
T,Y=RK.integrate(F,t,y,tStop,h)#need to integrate this is in order to use it
printSoln(T,Y,freq)
fig2=plt.figure()
plt.plot(Y[:,0],Y[:,1], label='Velocity against position for an anharmonic oscillator')#since second column in Y was dx/dt and first column was x.
plt.axis('equal')
plt.xlabel('x')
plt.ylabel('dx/dt')
plt.title ('Velocity of oscillator against its position')
def G(t2,y2):
G=numpy.zeros(2)
G[0]=y2[1]
G[1]=(-w**2)*(y2[0])#this is saying z'=-w^2x, setting up differential equation.
return G
y2=numpy.array([1,0])#initial conditions
## example7_11
from bulStoer import *
from numarray import array,zeros,Float64
from printSoln import *
def F(x,y):
F = zeros((2),type=Float64)
F[0] = y[1]
F[1] =(-y[1] - y[0]/0.45 + 9.0)/2.0
return F
H = 0.5
xStop = 10.0
x = 0.0
y = array([0.0, 0.0])
X,Y = bulStoer(F,x,y,xStop,H)
printSoln(X,Y,1)
raw_input("\nPress return to exit")
F = numpy.zeros(2)
F[0] = y[1]
F[1] = (-w**2) * (
y[0]**3) #this is saying z'=-w^2x^3, setting up differential equation.
return F
y = numpy.array([1, 0]) #initial conditions
w = 1
t = 0. #initial value of t
tStop = 20. #final value of t
h = 0.001 #h=size of steps
freq = 1
T, Y = RK.integrate(F, t, y, tStop,
h) #need to integrate this is in order to use it
printSoln(T, Y, freq)
plt.plot(T, Y[:, 0], 'r-', label='f(t,x)')
plt.xlabel('t')
plt.ylabel('x')
plt.title('Motion of oscillator from t=0 to 20')
plt.legend()
plt.show()
fig1 = plt.figure() #this is for when you want more than one graph.
y2 = numpy.array([10, 0])
T, Y2 = RK.integrate(F, t, y2, tStop, h)
plt.plot(T, Y2[:, 0], 'b-', label='f(t,x)')
plt.xlabel('t')
plt.ylabel('x')
plt.title(
def r(u):
xs, ys = integrate(F, xStart, initCond(u), xStop, h)
y = ys[len(ys) - 1]
r = y[1] - 1.0
return r
from run_kut4 import *
from ridder import *
from printSoln import *
xStart = 0
xStop = 2.0
h = 0.1
freq = 2
initCond = lambda u: np.array([0.0, u])
u1 = 1
u2 = 10
u = ridder(r, u1, u2)
xs, ys = integrate(F, xStart, initCond(u), xStop, h)
printSoln(xs, ys, freq)
plt.plot(xs, ys[:, 0], 'o-', xs, ys[:, 1], '-')
plt.xlabel('x')
plt.legend(['y', 'dy/dx'], loc=1)
plt.grid()
plt.show()
