CODEWITHSUNDEEP
matlab
Xiao Wang
Xiao Wang

Reputation: 13

Matlab backward Euler using fzero

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;

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?

Upvotes: 1

Views: 2158

Answers (1)

user3717023
user3717023

Reputation:

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);

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);

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)

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

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);

Upvotes: 1

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