Python数值求解微分方程：欧拉法、隐式欧拉及Runge-Kutta算法实现

凌晨两点半

发布时间：2026-10-16| 6人浏览过

原创

本文介绍如何使用Python数值求解形如[[wp:a0]]的微分方程。详细解析了欧拉法、隐式欧拉法、三阶及四阶Runge-Kutta法的实现原理与代码逻辑，并通过具体案例对比各算法的计算结果，为数值分析提供实用的编程参考。

Python求解微分方程概述

首先直接展示相关代码实现。

本文主要探讨如何求解形如的微分方程。

基础算法：欧拉法

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

改进算法：隐式欧拉法

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

高阶算法：三阶与四阶Runge-Kutta法

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

算法实现与结果对比

5. 结果可视化展示

此外，为了确保计算过程顺利执行，需要引入必要的绘图库支持。

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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