Iteration, and Plotting Error, PYTHON Midpoint Method, Numerical Analysis II

Question

I'm trying to figure out why my code is not iterating the values correctly.

I defined the function here:

    """ 
=================================================================
    Example 4:  IVP of ODE y' = -(y+1)(y+3)
            with intial value y(0) = -2
            Exact solution is y(t) = -3 + 2(1+e^-2t)^-1
            from t in [0,2]
            h = 0.2
            Exercise 5.4 No. 3c on page 291

    def exmp4_def_fn(tau,w):

    return (-1.0*((w + 1.0)*(w + 3.0)))

    def exmp4_sol(t):

    return (-3.0 + 2.0*(1.0 + np.exp((-2.0*t))**-1.0))

Then I try to plot the approximated solution vs the true solution and print the error with this code:

N4 = 10
a4 = 0.0
b4 = 2.0
ya4 = -2.0
#defining function and true solution of function #4
def_fn4 = exmp_fn.exmp4_def_fn
sol4 = exmp_fn.exmp4_sol


#run Euler's method from ODE_Approx_methods for example #4
(t4, w4) = ODE_Approx_methods.euler(def_fn4, a4, b4, ya4, N4)

#Exact solutions for comparison of example #3
z4 = sol4(t4)

#plot comparison of exact solution w(t) and approximation y(t), example 
4'
plt.figure(4)

print('Errors at time mesh points, Example #4: ')
print(np.abs(np.transpose(np.ravel(w4)-z4)))

plt.rcParams.update({'font.size': 20})
plt.plot(t4,z4, 'b-' , marker= 'o', linewidth=2)
plt.plot(t4,w4, 'c-', marker = '*', linewidth=2)

plt.xlabel('t4')
plt.ylabel('w(t4)')
plt.legend([' Exact Solution', 'Euler Approximation, Example #4'], loc 
= 
'lower right' )

plt.show()

Im getting these values:

print(w4):[-2.         -1.8        -1.608      -1.4387328  -1.30173697 
-1.19925122 -1.12749094 -1.07974536 -1.04911908 -1.02995398 
-1.01815184]

print(z4):[  1.           1.9836494    3.45108186   5.64023385   
8.90606485 13.7781122   21.04635276  31.88929354  48.06506039  
72.19646889 108.19630007]


print(np.abs(np.transpose(np.ravel(w4)-z4))):[  3. 3.7836494   
5.05908186   7.07896665  10.20780182
14.97736342  22.17384371  32.9690389   49.11417947  73.22642287
109.2144519 ]

When im supposed to get negative values, and a much much smaller error. anyone understand where i went wrong?

score 0 · Answer 1 · answered Feb 27 '18 at 19:00

Your brackets are wrong. The solution is y(t) = -3 + 2(1+e^-2t)^-1, but in the function containing the solution it says (-3.0 + 2.0*(1.0 + np.exp((-2.0*t))**-1.0)). In the latter you're only applying the ^-1 to the exponential instead of the sum. So if you change it to (-3.0 + 2.0*(1.0 + np.exp(-2.0*t))**-1.0), it should work fine.

Iteration, and Plotting Error, PYTHON Midpoint Method, Numerical Analysis II

1 Answers1