I just started to learn simulation about PDEs and encountered difficulties. How can I use method-of-lines to solve the following three variables and draw a graph?

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Yidan 2024-8-21

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评论： William Rose 2024-8-21

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Yidan 2024-8-21

编辑：Yidan 2024-8-21

Thank you very much for your reply again! !!

After the first question, I learned the method called method of lines and used pdepe again according to Bill Greene suggestion.

(1) In this problem with three variables, since f(x,t,u,∂u∂x)=0 is set, the value of q(x,t) does not affect the result according to theory, but I found that different values of q(x,t) have a great impact on the result, so I asked again.

The code about this question(picture) is :

function [c,f,s] = pdefun(x,t,u,dudx) % Equation to solve

c = [1; 1; 1];

f = [0; 0; 0];

s = [-0.6; 1.32; 10].*dudx+[-1; -1; 0].*u;

end

function u0 = pdeic(x) % Initial Conditions

u0 = [sin(pi*x); sin(pi*x) ; 0];

end

function [pl,ql,pr,qr] = pdebc(xl,ul,xr,ur,t) % Boundary Conditions

pl = [2.2*ul(2)+ ul(1); 0; 0];

ql = [0; 1; 1];

pr = [-ur(2)+0.19*ur(1)+ul(3); 0; ur(3)];

qr = [0; 1; 1];

end

x = linspace(0,1,100);

t = linspace(0,1,30);

m = 0;

sol = pdepe(m,@pdefun,@pdeic,@pdebc,x,t);

u1 = sol(:,:,1);

u2 = sol(:,:,2);

u3 = sol(:,:,3);

surf(x,t,u1)

title('u_(x,t)')

xlabel('Distance x')

ylabel('Time t')

If you choose different q, the result changes. I don't whether i made some mistakes.

(For example, if ql = [0; 1; 1];qr = [1; 1; 1]; The sesult changes)

(2) And I tried to use the method of lines on this three variables, and the error was the problem of array index. I think I haven't fully understood it yet, but it is difficult to find materials to study and code analysis, so I want to learn and understand by comparing the codes of different variables.

I sincerely thank you for your detailed answer.