How to use ode23s to solve heat equation

Question

Xiaoyan 2014-11-13

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评论： Torsten 2014-11-24

assume ut=D*uxx D is constant, x = [-5:0.1:5], with delta_x=0.1 initial condition u(x0,0) = derivative of x boundary condition if x=-5 or 5 u(x,t)=0

hence can i write ODE that satisfy u1 to u99? then use ODE23s solve heat equation with initial data u50=1 and uj=0 where j=/50?

thank you!!

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Answer 1

Torsten 2014-11-21

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If you are still interested in a solution for your problem:

Choose

u(x)=1/eps*max(1-abs(x/eps),0)

for eps=0.001, e.g. as initial condition for your Problem (this is an approximation to the Dirac Delta function).

Choose a very fine grid around x=0 such that the initial guess function is sufficiently resolved.

Then write the usual ODEs that result from

ut=D*uxx

and solve them using ODE15s.

If needed, I could also give you the analytical solution for comparison.

Best wishes

Torsten.

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Xiaoyan 2014-11-21

Many thanks Torsten,

Yeah, I would like to see the analytical solution for comparison

Xiaoyan

Torsten 2014-11-24

The analytical solution for the problem

ut=D*uxx

u(x,0)=delta(x)

u(-5,t)=u(5,t)=0

is given by

u(x,t)=sum_{n=-oo}^{n=+oo}1/sqrt(Pi*D*t)*(exp(-(x-20*n)^2/(4*D*t))-exp(-(x-20*n+10)^2/(4*D*t)))

Best wishes

Torsten.

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Answer 2

Orion 2014-11-13

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Xiaoyan,

You're problem is a partial differential equation (PDE), so ode23 won't work, it's meant to solves ordinary differential equation (ODE).

for this kind of problem, if you have it, use the PDE toolbox .

If you don't have it, then you're gonna have to code your own finite element method (such as finite difference). you could probably some opensource code on the web for those methods.

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Torsten 2014-11-13

Xiaoyan wants to apply the method of lines to the heat equation and solve the resulting system of ODEs using ODE23.

But I don't understand which Initial and boundary conditions he/she wants to impose.

Best wishes

Torsten.

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Answer 3

Xiaoyan 2014-11-13

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Thank you both, Torsten is right about my approach. for my question

Initial condition is: u(x,0)=δ(x) Boundary condition is: u(-5,t)=u(5,t)=0

I hope this make the question clearer.

Many thanks, Xiaoyan

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Answer 4

Torsten 2014-11-13

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I guess numerical methods are not adequate if Dirac-Delta function is used as initial condition.

You should look up the analytical solution for this problem.

Best wishes

Torsten.

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Answer 5

Zoltán Csáti 2014-11-13

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I would suggest an other way. A time-dependent differential equation can be solved not only by the method of lines, but also using the so-called Rothe's approach, ie. you discretize first in time and then is space. So use for example the forward Euler-method to prepare the semi-discrete equation that can be discretized by FDM/FEM or a spectral method.