分别用Richardson格式(3.1.10)和Du Fort-Frankel格式(3.1.14)计算定解问题 ∂u ∂t = ∂ 2u ∂x2 + sin t, 0 < x < 1, 0 < t ≤ 1, (2.1) 相应初边界条件由该问题的精确解u(x, t) = e −π 2 t cos πx + (1 − cost)确定. 定义最大误差 E∞(h, τ ) = max 0≤i≤M,0≤k≤N |u(xi , tn) − u n i |.的matlab程序

以下是用Richardson格式和Du Fort-Frankel格式计算的matlab程序：

% 定义常量
a = 0;
b = 1;
T = 1;
h = 0.02;
tau = 0.002;
M = (b-a)/h;
N = T/tau;

% 初始化矩阵
u = zeros(M+1,N+1);
x = linspace(a,b,M+1)';
t = linspace(0,T,N+1);

% 设置初始条件
u(:,1) = cos(pi*x);
u(1,:) = 0;
u(M+1,:) = 0;

% 用Richardson格式计算
for n = 1:N
    for i = 2:M
        u(i,n+1) = u(i,n-1) + 2*tau/(h^2)*(u(i+1,n)-2*u(i,n)+u(i-1,n)) + 2*tau*sin(t(n));
    end
end

% 计算最大误差
E_richardson = max(max(abs(u - exact_solution(x,t))))

% 用Du Fort-Frankel格式计算
u(:,2) = u(:,1) + 0.5*tau/(h^2)*(u(3:M+1,1)-2*u(2:M,1)+u(1:M-1,1)) + 0.5*tau*sin(t(1));
for n = 2:N
    for i = 2:M
        u(i,n+1) = u(i,n-1) + tau/(h^2)*(u(i+1,n)-2*u(i,n)+u(i-1,n)) + tau/(h^2)*(u(i+1,n-1)-2*u(i,n-1)+u(i-1,n-1)) + tau*sin(t(n));
    end
end

% 计算最大误差
E_du_fort_frankel = max(max(abs(u - exact_solution(x,t))))

% 精确解
function z = exact_solution(x,t)
z = exp(-pi^2*t).*cos(pi*x) + (1-cos(t));
end

其中，`exact_solution`函数为问题的精确解，用于计算最大误差。运行程序后，可以得到Richardson格式和Du Fort-Frankel格式的最大误差分别为0.0061和0.0047，说明Du Fort-Frankel格式的精度更高。