Python数值求解微分方程方法(欧拉法,隐式欧拉)

Roselani ·

更新时间:2024-11-13

· 1077 次阅读

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()

到此这篇关于Python数值求解微分方程方法(欧拉法,隐式欧拉)的文章就介绍到这了,更多相关Python数值求解微分方程内容请搜索软件开发网以前的文章或继续浏览下面的相关文章希望大家以后多多支持软件开发网！

微分方程微分欧拉方法 Python

1024 个赞