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non linear partial diferential equation

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sepideh
sepideh 2024-1-23
编辑： Torsten 2024-1-25
Hello
I need to insert this equation in MATLAB
would you please help me. I have no Idea
It may similar to heat equation, but there are some extra terms which I do not know how to deal with.

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Torsten
Torsten 2024-1-24
移动：Torsten 2024-1-24
I don't see a similarity to the heat equation.
You will have to discretize the left-hand side of the equation and use ode15s to solve for H in the grid points in X-direction.
Look up "method-of-lines" for more details.
"sign" is "sign" in MATLAB and "absolute value" is "abs" in MATLAB.
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sepideh
sepideh 2024-1-25
编辑：sepideh 2024-1-25
Dear Torsten
I searched about MOL, but still its not clear for me how to deal with That H which is multiplied into the sqaure root inside the derivative.
I saw several examples of Burger's equation with the form which had been solved by this method. I understand now what you mean by discretisizing L.H.S ,but I still dont know hnow to encounter H in those formulas.
Torsten
Torsten 2024-1-25
编辑：Torsten 2024-1-25
I don't know which discretization is optimal in your case, but you could consider approximating
d/dx (H * ( sqrt(1+4*N*abs(dH/dx)) - 1) ) @ X(i) as
{[H * ( sqrt(1+4*N*abs(dH/dx)) - 1)] @ X(i+1/2) - [H * ( sqrt(1+4*N*abs(dH/dx)) - 1)] @X(i-1/2)} / ( X(i+1/2)-X(i-1/2) ) ~
{[(H(i+1)+H(i))/2 * (sqrt(1+4*N*abs((H(i+1)-H(i))/(X(i+1)-X(i))))-1)] - ...
[(H(i)+H(i-1))/2 * (sqrt(1+4*N*abs((H(i)-H(i-1))/(X(i)-X(i-1))))-1)]} / ( X(i+1/2)-X(i-1/2) )
where
X(i+1/2) = (X(i+1)+X(i))/2
X(i-1/2) = (X(i)+X(i-1))/
and assuming that N is a constant.

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