Python数值求解微分方程(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()

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