Problem Sheet Question 2a

The general form of the population growth differential equation

(305)\[\begin{equation} y^{'}=y-t, \ \ (0 \leq t \leq 2) \end{equation}\]

with the initial condition

(306)\[\begin{equation}y(0)=2.\end{equation}\]

For N=4 with the analytic (exact) solution

(307)\[\begin{equation} y= e^{t}+t+1.\end{equation}\]

3-step Adams Bashforth

The 3-step Adams Bashforth difference equation is

(308)\[\begin{equation}w_{i+1} = w_{i} + \frac{h}{12}(23f(t_i,w_i)-16f(t_{i-1},w_{i-1})+5f(t_{i-2},w_{i-2})) \end{equation}\]

where

(309)\[\begin{equation}w_{i+1} = w_{i} + \frac{h}{12}(23(w_i-t_i)-(w_{i-1}-t_{i-1})+5(w_{i-2}-t_{i-2})). \end{equation}\]

import numpy as np
import math 

%matplotlib inline
import matplotlib.pyplot as plt # side-stepping mpl backend
import matplotlib.gridspec as gridspec # subplots
import warnings

warnings.filterwarnings("ignore")

def myfun_ty(t,y):
    return y-t



#PLOTS
def Adams_Bashforth_3step(N,IC):

    x_end=2
    x_start=0
    INTITIAL_CONDITION=IC
    h=x_end/(N)
    N=N+2;
    t=np.zeros(N)
    w=np.zeros(N)
   
    Analytic_Solution=np.zeros(N)
    k=0
    w[0]=INTITIAL_CONDITION
    Analytic_Solution[0]=INTITIAL_CONDITION
    t[0]=x_start
    t[1]=x_start+1*h
    t[2]=x_start+2*h
    w[1]=math.exp(t[1])+t[1]+1
    w[2]=math.exp(t[2])+t[2]+1
    Analytic_Solution[1]=math.exp(t[1])+t[1]+1
    Analytic_Solution[2]=math.exp(t[2])+t[1]+1

    for k in range (2,N-1):
        w[k+1]=w[k]+h/12.0*(23*myfun_ty(t[k],w[k])-16*myfun_ty(t[k-1],w[k-1])+5*myfun_ty(t[k-2],w[k-2]))
        t[k+1]=t[k]+h
        Analytic_Solution[k+1]=2*math.exp(t[k+1])+t[k+1]+1

    fig = plt.figure(figsize=(10,4))
    # --- left hand plot
    ax = fig.add_subplot(1,3,1)
    plt.plot(t,w,'o:',color='blue')
    #ax.legend(loc='best')
    plt.title('Numerical Solution h=%s'%(h))

    # --- right hand plot
    ax = fig.add_subplot(1,3,2)
    plt.plot(t,Analytic_Solution,color='black')
    plt.title('Analytic Solution')

   
    ax = fig.add_subplot(1,3,3)
    plt.plot(t,Analytic_Solution-w,'o:',color='red')
    plt.title('Error')
 # --- title, explanatory text and save



    # --- title, explanatory text and save
    fig.suptitle(r"$y'=t-y$", fontsize=20)
    plt.tight_layout()
    plt.subplots_adjust(top=0.85)    
    print(t)
    print(Analytic_Solution)
    print(w)

Adams_Bashforth_3step(4,2)

[0.  0.5 1.  1.5 2.  2.5]
[ 2.          3.14872127  4.21828183 11.46337814 17.7781122  27.86498792]
[ 2.          3.14872127  4.71828183  6.93248773 10.21158419 15.2340026 ]