椭圆型偏微分方程五点差分法python

椭圆型偏微分方程的五点差分法是一种常用的数值解法。下面是一个使用Python实现的例子：

import numpy as np

def solve_elliptic_equation(f, g, h, nx, ny, Lx, Ly):
    dx = Lx / (nx - 1)
    dy = Ly / (ny - 1)

    x = np.linspace(0, Lx, nx)
    y = np.linspace(0, Ly, ny)

    # 初始化解向量
    u = np.zeros((nx, ny))

    # 设置边界条件
    u[0, :] = g(x[0], y)
    u[-1, :] = g(x[-1], y)
    u[:, 0] = h(x, y[0])
    u[:, -1] = h(x, y[-1])

    # 迭代求解
    for i in range(1, nx - 1):
        for j in range(1, ny - 1):
            u[i, j] = (u[i-1, j] + u[i+1, j] + u[i, j-1] + u[i, j+1] - dx**2 * f(x[i], y[j])) / 4

    return u

# 定义椭圆型偏微分方程的参数和边界条件
def f(x, y):
    return 0

def g(x, y):
    return 0

def h(x, y):
    return np.sin(np.pi * x) * np.sin(np.pi * y)

# 调用函数求解
nx = 101
ny = 101
Lx = 1
Ly = 1
u = solve_elliptic_equation(f, g, h, nx, ny, Lx, Ly)

# 打印结果
print(u)

这段代码使用了NumPy库来进行数组操作和数值计算。首先定义了一个`solve_elliptic_equation`函数，该函数接受椭圆型偏微分方程的参数和边界条件，并返回解向量。然后定义了椭圆型偏微分方程的参数和边界条件函数`f(x, y)`、`g(x, y)`和`h(x, y)`。最后调用`solve_elliptic_equation`函数求解，并打印结果。