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sajar February 28, 2004 20:03

exact solution of burger's equation
 
can anyone help me getting exact solution of inviscid burger's equation?

rvndr February 29, 2004 01:05

Re: exact solution of burger's equation
 
For u_t + u u_x =0

we get

u(x,t)=f(x - u(x,t)t) for any f(x). which gives us an implicit solution for u(x,t) but this solution can be used to obtain a slope in order to check the change of u wrt "x".

Hope this helps

sajr February 29, 2004 21:03

Re: exact solution of burger's equation
 
hey rvndr i did not get exactly what you write. can u plz elaborate.

Tom March 1, 2004 06:12

Re: exact solution of burger's equation
 
The characteristics of the Burger equation satisfy

dt/ds = 1, dx/ds = u, du/ds = 0

where s is a parameter along the characteristic. Initially (s=0) set

t = 0, x= z, u=f(z),

then we have (t=s)

u=f(z), x = z + tf(z)

Tom.

ag March 1, 2004 10:07

Re: exact solution of burger's equation
 
You can also solve it using a Cole-Hopf transformation. Try doing a search on google for Cole-Hopf and burger's equation.

Tom March 1, 2004 10:32

Re: exact solution of burger's equation
 
The Hopf-Cole transformation is for the viscous problem and doesn't work in the inviscid case - unless you are suggesting solving the viscous problem exactly and then taking the limit of zero viscosity.

rvndr March 2, 2004 00:52

Re: exact solution of burger's equation
 
hi sajar,

Tom explained it nicely. Still if you have doubt post that PDE and let me solve that euation for you if I can.

rvndr

rvndr March 2, 2004 00:57

Re: exact solution of burger's equation
 
hi sajar,

Tom explained it nicely. Still if you have doubt post that PDE for which you want to know the exact solution and let me solve that euation for you if I can.

rvndr

Saverio March 3, 2004 22:47

Re: exact solution of burger's equation
 
The viscous and inviscid Burger's equations display markedly different dynamics in the shock region - are you sure that you can get to the inviscid solution with such a limiting process? I would guess not, but I haven't actually done it.

Tom March 4, 2004 05:55

Re: exact solution of burger's equation
 
Yes you can - it's simple matched asymptotic expansions (see the book by Kevorkian and Cole and also Whitham's book on linear and nonlinear waves). Basically this is how shock capturing methods work in numerical models.


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