例題 5-4 利用 二階 Runge-Kutta 解一階常微分方程式 y' = - y + t +1 , y(0)=1 , 0<= t <= 1 取 h=0.2 真實解 W(t) = exp(-t) + t

'''
#例題 5-4 利用 二階 Runge-Kutta  解一階常微分方程式
# y' = - y + t  +1  , y(0)=1  , 0<= t <= 1
# 取 h=0.2
# 真實解 W(t) =   exp(-t)  + t

 * Second Order Runge-Kutta 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========
n=5;
a=0.0
b=1.0
y0=1.0
h=(b-a)/n
y=y0
t0=a
t=t0
print("t \t\t y(t) \t\t\t   w(t)=exp(-t)+t \t\t  error");
print("=========================================================")
print("{%.2f} \t\t  {%10.7f}  \t\t  {%10.7f} \t\t {%10.7f} " %(t,y,W(t),abs(y-W(t)) ) )
for i in range (1,n+1):
    k1=h*F(y,t)
    k2=h*F((y+k1),(t+h))
    y=y+0.5*(k1+k2)
    t=t+h
    #print(t,W(t))
    print("{%.2f} \t\t  {%10.7f}  \t\t  {%10.7f} \t\t {%10.7f} " %(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/EX5-4.py ============
t                    y(t)             w(t)=exp(-t)+t     error
=========================================================
{0.00}   { 1.0000000}    { 1.0000000} { 0.0000000} 
{0.20}   { 1.0200000}    { 1.0187308} { 0.0012692} 
{0.40}   { 1.0724000}    { 1.0703200} { 0.0020800} 
{0.60}   { 1.1513680}    { 1.1488116} { 0.0025564} 
{0.80}   { 1.2521218}    { 1.2493290} { 0.0027928} 
{1.00}   { 1.3707398}    { 1.3678794} { 0.0028604} 
>>>