微分方程数值解用matlab练习

Posted 2020-12-08 lwqq3

实习题1：

将xx化成一阶方程组，并用Euler法和改进的Euler法求解，步长h = 0.1, 0.05。

画出Euler法，Euler改进法以及精确解的数值结果图形

clear all;
clc;
h1 = 0.05;
len = 1 + 1 / h1;
u = zeros(1, len);
v = zeros(1, len);
t = zeros(1, len);
exact = zeros(1, len);

u(1) = 0;
v(1) = 1;
t(1) = 0;
exact(1) = 0;
for i = 1:len-1
    t(i + 1) = i * h1;
    u(i + 1) = u(i) + h1 * v(i);
    v(i + 1) = v(i) - h1 * u(i);
    exact(i + 1) = sin(h1 * i);
end

u1 = zeros(1, len);
v1 = zeros(1, len);
u1(1) = 0;
v1(1) = 1;
for i = 1:len-1
    u1(i + 1) = ((1 + h1 * h1 / 4) * u(i) + h1 * v(i)) / (1 - h1 * h1 / 4);
    v1(i + 1) = v1(i) + (u1(i) + u1(i + 1)) * h1 / 2;
end
    
plot(t, u, ‘r*‘, ‘markersize‘, 10);
hold on
plot(t, u1, ‘r.‘, ‘markersize‘, 20);
hold on
plot(t, exact);
grid on

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