def adaptive_step_demo(width, height, font):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
options = {'Refine': 1}
def dydt(t, y):
return [-1 * y[0] + 10 * np.exp(-1 * (t - 2) * (t - 2) / 0.002)]
r = ode45(dydt, [1, 4], [0.5], options)
r2 = ode45(dydt, [1, 4], [0.5])
plt.plot(r2.tout, r2.yout[0])
plt.vlines(r.tout,
ymin=0,
ymax=r.yout[0],
colors='b',
linestyles='dashed',
alpha=0.5)
plt.legend(['y(t)', 'Steps'])
plt.title(
'Solution for y\'(t) = -y(t) + 10e^(-(t-2)^2/0.002) for t = {1,4}')
plt.xlabel('t')
plt.ylabel('y')
def multiple_initial_conditions_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [
y[0] * math.cos(t), y[1] * math.cos(t), y[2] * math.cos(t),
y[3] * math.cos(t), y[4] * math.cos(t)
]
tspan = [0, 10]
y0 = [1, 2, 3, 4, 5]
r = ode45(dydt, tspan, y0)
plt.plot(r.tout, r.yout[0])
plt.plot(r.tout, r.yout[1])
plt.plot(r.tout, r.yout[2])
plt.plot(r.tout, r.yout[3])
plt.plot(r.tout, r.yout[4])
plt.title(
'Solution for y\'(t) = y * cos(t) with y(0) = 1,2,3,4,5 for t = {0,10}'
)
plt.xlabel('t')
plt.ylabel('y')
def lorenz_demo():
fig = plt.figure()
fig.set_size_inches(15, 12)
fig.patch.set_alpha(1)
ax = fig.gca(projection='3d')
sigma = 10.
rho = 28.
beta = 8. / 3.
x0 = [0, 1, 2]
dt = 0.001
def lorenz(t, y):
A = np.array([[-1 * beta, 0, y[1]], [0, -1 * sigma, sigma],
[-1 * y[1], rho, -1]])
y = np.transpose(np.array([y]))
return np.transpose(np.matmul(A, y))[0]
tspan = np.linspace(dt, 50, 50000)
options = {'RelTol': 1e-12, 'AbsTol': 1e-12}
r = ode45(lorenz, tspan, x0, options)
ax.plot(r.yout[0], r.yout[1], r.yout[2])
ax.patch.set_facecolor('white')
ax.grid(False)
plt.show()
def cosbasic2(self,result):
def f(t,y):
return [math.cos(t),2*math.cos(t)]
opt={'AbsTol':[1e-2,2e-7],'RelTol':1e-6}
tspan = [8000000000000,8000000000010]
y0 = [1, 0]
sol=ode45(f,tspan,y0,opt)
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
def trigtol(self,result):
def f(t,y):
return [y[0]*np.cos(t),y[1]*np.sin(t)]
opt={'RelTol' : 0.0009,'AbsTol' : 3.e-8,'NormControl' : 'on','Refine' : 3,'NonNegative' : [0, 1]}
Tspan = [19, 53]
Y0 = [6, 38]
sol=ode45(f,Tspan,Y0,opt)
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
def cosbasic3(self,result):
def f(t,y):
return [math.cos(t),2*math.cos(t)]
opt={'AbsTol':[1e-2,2e-7],'RelTol':1e-6,'InitialStep':0.2}
tspan = np.linspace(0,10,21)
y0 = [1, 0]
sol=ode45(f,tspan,y0,opt)
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
def cosbasic1(self,result):
def f(t,y):
return [math.cos(t),2*math.cos(t)]
mass = np.array([[0.46,0.16],[6.4,0.11]])
opt={'Mass' : mass, 'NonNegative' : [0]}
Tspan = [8, 59]
Y0 =[84, 0.9]
sol=ode45(f,Tspan,Y0,opt)
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
def polyNN(self,result):
tspan = [-6,5]
y0=[25,50,25,50]
opt={'NonNegative':[0,1]}
def f(t,y):
x=0.02*(3*pow(t,5)-62*pow(t,3)+42*pow(t,2)+45*t+18)
return [x,x,x,x]
sol = ode45(f,tspan,y0,opt)
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
def absolute_tolerance_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [math.cos(t), math.sin(t)]
option = {'AbsTol': [2.1e-6, 4.5e-8]}
r = ode45(dydt, [0, 10], [1, 0], option)
plt.plot(r.tout, r.yout[0])
plt.plot(r.tout, r.yout[1])
def refine_demo_1(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [np.cos(t), np.sin(t)]
options = {'Refine': 20}
r = ode45(dydt, [0, 10], [1, 0], options)
plt.plot(r.tout, r.yout[0], 'b', label='refine = 1')
options = {'Refine': 1}
r2 = ode45(dydt, [0, 10], [1, 0], options)
plt.plot(r2.tout, r2.yout[0], 'r', label='refine = 10')
plt.plot(r.tout, r.yout[1], 'b', label='refine = 1')
plt.plot(r2.tout, r2.yout[1], 'r', label='refine = 10')
plt.title('Refinement demo')
plt.xlabel('t')
plt.ylabel('y')
plt.legend(['refine = 20', 'refine = 1'])
def ball(self,result):
y0 = [0,20]
tspan = [0,30]
def f(t,y):
return [y[1], -9.8]
def events(t,y):
return [y[0]],[1],[-1]
opt = {'Events':events}
sol = ode45(f,tspan,y0,opt)
teout,yeout,ieout=sol.get_events()
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
self.compare_events(result,teout,yeout,ieout)
def trigevents(self,result):
y0 = [0,-1]
tspan = [0,30]
def f(t,y):
return [np.cos(t),np.sin(t)]
def events(t,y):
return [y[0],y[1],y[0]],[0,0,0],[-1,0,1]
opt = {'Events':events}
sol = ode45(f,tspan,y0,opt)
teout,yeout,ieout=sol.get_events()
self.compare_ty(result,sol.tout,sol.yout)
self.compare_stats(result,sol.get_stats())
self.compare_events(result,teout,yeout,ieout)
def bouncing_ball_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def events(t, y):
value = [y[0]]
isterminal = [1]
direction = [-1]
return value, isterminal, direction
options = {'Events': events, 'NormControl': 'on', 'Refine': 10}
tspan = [0, 30]
tstart = 0
y0 = [0, 20]
def dydt(t, y):
return [y[1], -9.8]
t = np.array([])
y = np.array([])
te = np.array([])
ye = np.array([])
for i in range(10):
tspan = [tstart, 30]
r = ode45(dydt, tspan, y0, options)
n = len(r.tout)
y0[1] = 0.9 * y0[1]
t = np.append(t, r.tout)
y = np.append(y, r.yout[0])
te = np.append(te, r.teout)
ye = np.append(ye, r.yeout[0])
tstart = r.tout[n - 1]
plt.plot(t, y)
plt.scatter(te, ye, color='r')
plt.title('Bouncing Ball Trajectory')
plt.xlabel('Time (s)')
plt.ylabel('Height (cm)')
def simple_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [np.cos(t)]
tspan = [0, 10]
y0 = [1]
r = ode45(dydt, tspan, y0)
plt.plot(r.tout, r.yout[0])
plt.title('Solution for y\'(t) = cos(t) with y(0) = 1 for t = {0,10}')
plt.xlabel('t')
plt.ylabel('y')
def choosen_points_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [y[0] * np.cos(t), y[1] * np.cos(t), y[2] * np.cos(t)]
tspan = np.linspace(0, 10, num=101)
y0 = [1, 2, 3]
r = ode45(dydt, tspan, y0)
plt.plot(r.tout, r.yout[0])
plt.plot(r.tout, r.yout[1])
plt.plot(r.tout, r.yout[2])
plt.title('Trigonometric functions with chosen points')
plt.xlabel('t')
plt.ylabel('y')
def orbit_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
mu = 1 / 82.45
mustar = 1 - mu
r13 = math.pow(((y[0] + mu) * (y[0] + mu) + y[1] * y[1]), 1.5)
r23 = math.pow(((y[0] - mustar) * (y[0] - mustar) + y[1] * y[1]), 1.5)
x1 = y[2]
x2 = y[3]
x3 = 2 * y[3] + y[0] - mustar * ((y[0] + mu) / r13) - mu * (
(y[0] - mustar) / r23)
x4 = -2 * y[2] + y[1] - mustar * (y[1] / r13) - mu * (y[1] / r23)
return [x1, x2, x3, x4]
def events(t, y):
y0 = [1.2, 0, 0, -1.04935750983031990726]
dDSQdt = 2 * (((y[0] - y0[0]) * y[2]) + ((y[1] - y0[1]) * y[3]))
value = [dDSQdt, dDSQdt]
isterminal = [1, 0]
direction = [1, -1]
return [value, isterminal, direction]
y0 = [1.2, 0, 0, -1.04935750983031990726]
tspan = [0, 7]
options = {'Events': events, 'RelTol': 1e-5, 'AbsTol': 1e-4}
r = ode45(dydt, tspan, y0, options)
plt.plot(r.yout[0], r.yout[1])
plt.scatter(r.yeout[0], r.yeout[1], color='r')
plt.title('Restricted three body problem')
plt.xlabel('x')
plt.ylabel('y')
def nonnegative_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [-1, -t, -t * t]
options = {'NonNegative': [0, 1, 2]}
tspan = [0, 3]
y0 = [1, 2, 3]
r = ode45(dydt, tspan, y0, options)
plt.plot(r.tout, r.yout[0], label='y\'(t) = -1')
plt.plot(r.tout, r.yout[1], label='y\'(t) = -t')
plt.plot(r.tout, r.yout[2], label='y\'(t) = -t^2')
plt.legend()
plt.title('Non-Negative solutions')
plt.xlabel('t')
plt.ylabel('y')
def mass_demo(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def mass(t):
return [[t + 1, -1], [1, t + 1]]
def dydt(t, y):
return [math.cos(t), 1]
options = {'Mass': mass, 'MStateDependence': 'none'}
y0 = [0, 0]
tspan = [0, 10]
r = ode45(dydt, tspan, y0, options)
plt.plot(r.tout, r.yout[0])
plt.plot(r.tout, r.yout[1])
plt.title('Mass demo')
plt.xlabel('t')
plt.ylabel('y')
def solve_ode(inputs, result):
sol = ode45(inputs.fun, inputs.tspan, inputs.y0, inputs.get_options(),
inputs.varargin)
result.compare_ty(sol.get_t(), sol.get_y())
result.compare_stats(sol.get_stats())
def refine_demo_2(width=8, height=6, font=12):
plt.figure()
fig = plt.gcf()
fig.set_size_inches(width, height)
plt.rcParams.update({'font.size': font})
def dydt(t, y):
return [np.cos(t)]
options = {'Refine': 4}
r = ode45(dydt, [0, 5], [1], options)
plt.plot(r.tout[12:21], r.yout[0, 12:21], 'b')
options = {'Refine': 1}
r2 = ode45(dydt, [0, 5], [1], options)
plt.plot(r2.tout[3:6], r2.yout[0, 3:6], 'r')
plt.vlines(x=1.2009509145207664,
ymin=1.8,
ymax=1.9323832558773466,
colors='r',
linestyles='dashed',
alpha=0.7)
plt.vlines(x=1.3259509145207664,
ymin=1.8,
ymax=1.970175241884764,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=1.4509509145207664,
ymin=1.8,
ymax=1.9928273192095607,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=1.5759509145207664,
ymin=1.8,
ymax=1.9999866178575545,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=1.7009509145207664,
ymin=1.8,
ymax=1.9915418852636042,
colors='r',
linestyles='dashed',
alpha=0.7)
plt.vlines(x=1.8259509145207664,
ymin=1.8,
ymax=1.9676234949145097,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=1.9509509145207664,
ymin=1.8,
ymax=1.9286072692934104,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=2.0759509145207664,
ymin=1.8,
ymax=1.8751005881572886,
colors='b',
linestyles='dashed',
alpha=0.5)
plt.vlines(x=2.2009509145207664,
ymin=1.8,
ymax=1.8079364922709038,
colors='r',
linestyles='dashed',
alpha=0.7)
plt.xticks([1.2009509145207664, 1.7009509145207664, 2.2009509145207664])
plt.title('Solution for y\'(t) = cos(t) for t = {0,5}')
plt.xlabel('t')
plt.ylabel('y')
plt.legend(['refine = 4', 'refine = 1', 'Step', 'Interpolation'])