Python数值求解微分方程方法(欧拉法,隐式欧拉)

不说什么,先上代码

这里先求解形如的微分方程

1.欧拉法

def eluer(rangee,h,fun,x0,y0):
   step = int(rangee/h)
   x = [x0] + [h * i for i in range(step)]
   u = [y0] + [0     for i in range(step)]
   for i in range(step):
       u[i+1] = u[i] + h * fun(x[i],u[i])
   plt.plot(x,u,label = "eluer")
   return u

2.隐式欧拉法

def implicit_euler(rangee,h,fun,x0,y0):
   step = int(rangee/h)
   x = [x0] + [h * i for i in range(step)]
   u = [y0] + [0     for i in range(step)]
   v = ["null"] + [0 for i in range(step)]
   for i in range(step):
           v[i+1] = u[i] + h * fun(x[i],u[i])
           u[i+1] = u[i] + h/2 * (fun(x[i],u[i]) + fun(x[i],v[i+1]))
   plt.plot(x,u,label = "implicit eluer")
   return u

3.三阶runge-kutta法

def order_3_runge_kutta(rangee,h,fun,x0,y0):
   step = int(rangee/h)
   k1,k2,k3 = [[0 for i in range(step)] for i in range(3)]
   x = [x0] + [h * i for i in range(step)]
   y = [y0] + [0     for i in range(step)]
   for i in range(step):
       k1[i] = fun(x[i],y[i])
       k2[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k1[i])
       k3[i] = fun(x[i]+0.5*h,y[i]+2*h*k2[i]-h*k1[i])
       y[i+1] = y[i] + 1/6 * h * (k1[i]+4*k2[i]+k3[i])
   plt.plot(x,y,label = "order_3_runge_kutta")
   return y

4.四阶runge-kutta法

def order_4_runge_kutta(rangee,h,fun,x0,y0):
   step = int(rangee/h)
   k1,k2,k3,k4 = [[0 for i in range(step)] for i in range(4)]
   x = [x0] + [h * i for i in range(step)]
   y = [y0] + [0     for i in range(step)]
   for i in range(step):
       k1[i] = fun(x[i],y[i])
       k2[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k1[i])
       k3[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k2[i])
       k4[i] = fun(x[i]+h,y[i]+h*k3[i])
       y[i+1] = y[i] + 1/6 * h * (k1[i]+2*k2[i]+2*k3[i]+k4[i])
   plt.plot(x,y,label = "order_4_runge_kutta")
   return y

5.上图

 当然,想要成功操作,得加上这个

rangee = 1
fun = lambda x,y:y-2*x/y

implicit_euler(rangee,0.0001,fun,0,1)
order_4_runge_kutta(rangee,0.0001,fun,0,1)
order_3_runge_kutta(rangee,0.0001,fun,0,1)
eluer(rangee,0.0001,fun,0,1)
plt.legend()
plt.show()

来源：https://blog.csdn.net/Chandler_river/article/details/124295877