二维椭圆形方程五点差分python

二维椭圆形方程是一个偏微分方程，可以用来描述许多物理问题，如电场和热传导等。五点差分法是一种常见的数值求解方法，用于解决偏微分方程。以下是二维椭圆形方程五点差分的Python代码实现：

import numpy as np

# 定义椭圆形方程
def f(x, y):
    return 2*x**2 + 3*y**2

# 定义边界条件
def boundary_condition(x, y):
    return 0

# 定义求解函数
def solve(f, boundary_condition, n=100, m=100, tol=1e-6, max_iter=10000):
    h = 1/n
    k = 1/m
    x = np.linspace(0, 1, n+1)
    y = np.linspace(0, 1, m+1)
    # 初始化
    u = np.zeros((n+1, m+1))
    for i in range(n+1):
        u[i, 0] = boundary_condition(x[i], 0)
        u[i, m] = boundary_condition(x[i], 1)
    for j in range(m+1):
        u[0, j] = boundary_condition(0, y[j])
        u[n, j] = boundary_condition(1, y[j])
    # 迭代求解
    for k in range(max_iter):
        u_old = u.copy()
        for i in range(1, n):
            for j in range(1, m):
                u[i, j] = (u_old[i-1, j] + u_old[i+1, j] + u_old[i, j-1] + u_old[i, j+1] - h**2*f(x[i], y[j]))/4
        # 判断收敛
        if np.max(np.abs(u-u_old)) < tol:
            break
    return u

# 求解并输出结果
u = solve(f, boundary_condition)
print(u)