Matlab使用fzero向后欧拉 [英] Matlab backward Euler using fzero

问题描述

我正在尝试整合dy3dt = @(t，y)-41 *(y-t)+ 1;使用后向Euler在h = 0.05时超过[0，1].初始条件y(0)为1.

I am trying to integrate dy3dt = @(t,y) -41*(y - t) + 1; over [0, 1] with h = 0.05 using backward Euler. The initial condition, y(0) is 1.

yvals3 = [1]; %inital condition
dy3dt = @(t,y) -41*(y - t) + 1;
for t = 0:0.05:0.95; 

    index = round(t*20 + 1); %convert t into index of the last step
    oldy = yvals3(index);
    newy1 = oldy + h.*dy3dt(t, oldy);
    newy2 = oldy + h.*dy3dt(t+ 0.05, newy1); 
    yvals3 = [yvals3; newy2];
end
tvals3 = 0:0.05:1;

当我绘制tvals3和yvals3的图形时，我得到了一个不寻常的指数图，所以我的方法可能不正确.关于如何使用fzero实现此功能的任何见解?

When I graph tvals3 and yvals3, I am getting an unusual exponential graph, so my approach is probably incorrect. Any insight on how to implement this using fzero?

推荐答案

通常的(向前)欧拉方法可以表示为从切线上的已知点开始并获得新点:

The usual (forward) Euler's method can be expressed as going from a known point on a tangent, and getting new point:

newy = oldy + tstep*dydt(oldt, oldy); 

向后Euler方法向后进行所有操作:它从切线上的一个新点(至今未知)开始向后，并到达了旧点:

The backward Euler method does everything backwards: it goes from a new (yet unknown) point on a tangent, backward, and hits the old point:

oldy = newy - tstep*dydt(newt, newy); 

最后一行不是可以执行的命令，因为newy是未知的.相反，这是一个需要解决的方程式:

This last line is not a command that can be executed, since newy is unknown. Instead, it's an equation to be solved:

newy = fzero(@(y) y - tstep*dydt(newt,y) - oldy, oldy)

(第二个参数oldy是对fzero所需的根的初始猜测.)

(The second parameter, oldy, is an initial guess to the root, which fzero needs.)

这是在您的设置中可以使用的方式.我写的代码可能不是最高效的Matlab，但是(希望)可以使计算逻辑清晰明了.

Here's how this could work in your setup. I wrote the code in a way that may not be the most Matlab-efficient, but (I hope) makes the logic of computation clear.

dydt = @(t,y) -41*(y - t) + 1;   % right hand side
tvals = [0];                     % initial t
yvals = [1];                     % initial y
tstep = 0.05;                    % time step 
numsteps = 20;                   % number of steps to go
for i=1:numsteps; 
    oldt = tvals(end);           % pick the last entry of array
    oldy = yvals(end);
    newt = oldt + tstep;         % step forward in t 
    newy = fzero(@(y) y - tstep*dydt(newt,y) - oldy, oldy);  % backward Euler 
    tvals = [tvals; newt];       
    yvals = [yvals; newy];
end
plot(tvals, yvals);

这篇关于Matlab使用fzero向后欧拉的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！