例題 5-1 利用向前 Euler 解 一階常微分方程式 y' = - y + t +1 , y(0)=1 , 0<= t <= 1 取 h=0.02 真實解 W(t) = exp(-t) + t

'''
#例題 5-1 利用向前 Euler 解 一階常微分方程式
# y' = - y + t +1  , y(0)=1  , 0<= t <= 1
# 取 h=0.02
# 真實解 W(t) =  exp(-t)  + t
 * Forward Euler Method is used for
 * solving y'=f(y,t) of first order  ordinary differential equation
 * with initial condition y(0)=y0  known.
 *
 '''
import math
def F(y,t):
    return ( -y+ t +1 )

def w(t):
    return ( math.exp(-t) + t)

#======== main========
t0=0
t1=1
h=0.02
n=int ((t1-t0)/h)
y0= 1

y=y0;
t=t0;
print("t \t\t y(t) \t\t\t       w(t) \t\t\t  error");
print("=========================================================")
print("{%.2f} \t\t  {%10.6f}  \t\t  {%10.6f} \t\t {%10.6f} " %(t,y,w(t),abs(y-w(t)) ) )
for i in range (1,n+1):
    t=t+h
    y=y+h*F(y,t)
    if (i%10==0):
        print("{%.2f} \t\t  {%10.6f}  \t\t  {%10.6f} \t\t {%10.6f} " %(t,y,w(t),abs(y-w(t)) ) )

========= RESTART: F:/2018-09勤益科大數值分析/數值分析/PYTHON/EX5-1.py ============
t y(t)        w(t)   error
=========================================================
{0.00}   {  1.000000}    {  1.000000} {  0.000000} 
{0.20}   {  1.020731}    {  1.018731} {  0.002001} 
{0.40}   {  1.074256}    {  1.070320} {  0.003936} 
{0.60}   {  1.154575}    {  1.148812} {  0.005763} 
{0.80}   {  1.256786}    {  1.249329} {  0.007457} 
{1.00}   {  1.376886}    {  1.367879} {  0.009007} 
>>>