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I'm trying to plot the solution to a PDE, which is an infinite series:

So far this is my code, I can't see what's wrong with it. I've attached the graph i got below.

edit: the sin part is part of the summation, and the original pde is:

xRange = 0:0.1:6;

tRange = 0:0.1:6;

[x,t] = meshgrid(xRange,tRange);

u = zeros(size(x));

for n=1:300

u = u+(4/(n^3*pi^3)+(4/(n*pi)-4/(n^3*pi^3))*exp(-n^2*pi^2.*t)).*sin(n*pi/2*x);

end

surf(x,t,u);

xlabel("X");

ylabel("T");

zlabel("U");

John D'Errico
on 10 Jun 2021

Edited: John D'Errico
on 10 Jun 2021

Do you understand that the series you show is not the same as what you wrote in code?

Where did the sin term go in your code? Without it, you get something VERY different.

Edit:

ARGH. We seem to be chasing a moving problem.

Along at least one edge, you have a problem that the series is poorly convergent, and taking only 10 terms is inadequate. You would need many more terms in the series, when t==0. In fact, as I look, your series, it is divergent when t == 0. Why is that the case? When t == 0, the exponential goes away. So now you have an infinite sum of the terms

4/(n*pi)

And since this sub-series is divergent, you have a problem along that edge.

John D'Errico
on 10 Jun 2021

And I just updated my answer. Again, THINK about how that series reduces, when t == 0, or even when t is small. For small t, the series will be VERY slowly convergent. When t == 0, the series is divergent. And that means you get garbge when t == 0. Even for small t, taking only 10 terms is rediculously inadequate.

So again, substitute t == 0 into that series. DO IT NOW! Look at what you see. Do you see the sum from 1 to infinity of 4/pi*(1/n) as a subseries?

Do you understand that this series is divergent, that it becomes infinitely large? In mathematics, this is called the Harmonic series. Read here:

Do you see that it is divergent? And that means, when t = 0, or even for small t, you will get complete and utter garbage in your plots.

If you don't believe me, LOOK AT THE PLOT YOU PRODUCED. Do you see that in your plot? Look at the edge where t == 0. Even when t is small, you see strange stuff. That happens because when t is small, the result will be slowly convergent. When t is zero, it will be divergent.

Max Heiken
on 10 Jun 2021

Edited: Max Heiken
on 10 Jun 2021

The part with the sin is missing in your code.

for n=1:10

u = u+(4/(n^3*pi^3)+(4/(n*pi)-4/(n^3*pi^3))*exp(-n^2*pi^2*t)).*sin(n*pi*x/2);

end

surf(x,t,u);

edited: It seems like the sin part is supposed to be part of the summation.

KSSV
on 10 Jun 2021

MAy be you are looking for this:

n=1:10 ;

tRange = 0:0.05:1;

[n,t] = meshgrid(n,tRange);

u = (4./(n.^3*pi^3)+(4./(n*pi)-4./(n.^3*pi^3)).*exp(-n.^2*pi^2.*t));

surf(n,t,u);

xlabel("X");

ylabel("T");

zlabel("U");

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