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Backward Heat Equation with Transversality Condition

1

0

$begingroup$

I have a backward parabolic equation of the form:

beginequation
W_eta + aW_xx - bW = 0
endequation

s.t.

beginequation
lim_eta rightarrow infty(x,eta) = g(x)
endequation

were $x in mathbbR$, $eta geqslant 0$, and $a,b$ are positive constants.

Applying the following transformations:

beginalign
W(x,eta) &= U(x,t)e^beta \
t &= aeta
endalign

we would get the backward heat equation below

beginequation
U_t = - U_xx
endequation

However, the transversality condition becomes a problem, since as $eta rightarrow infty$, $e^beta rightarrow infty$.

Usually, if the terminal condition is of the form

beginequation
W(x,H) = g(x)
endequation

with $H$ finite, we could "reverse" it, that is, we could apply the following transformation:

beginequation
nu = H - eta
endequation

to obtain

beginequation
-W_nu + aW_xx - bW = 0
endequation

s.t.

beginequation
W(x,0) = g(x)
endequation

which we can solve the traditional way (Fourier transform). However, as my terminal condition happens only at infinity I can't apply the reverse transformation above, thus I don't know how to overcome this problem. Any hint or reference?

pde heat-equation

share|cite|improve this question

edited Apr 2 at 14:56

Nicolas Pimentel de Souza

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you tried solving the equation with $U(x,T)=g(x)$ and let $Ttoinfty$?
  $endgroup$
  – Dylan
  Apr 2 at 7:23
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have edited the question, now it reflects properly the problem I have.
  $endgroup$
  – Nicolas Pimentel de Souza
  Apr 2 at 14:48
  

  
  
  
  

add a comment | 

1

0

$begingroup$

I have a backward parabolic equation of the form:

beginequation
W_eta + aW_xx - bW = 0
endequation

s.t.

beginequation
lim_eta rightarrow infty(x,eta) = g(x)
endequation

were $x in mathbbR$, $eta geqslant 0$, and $a,b$ are positive constants.

Applying the following transformations:

beginalign
W(x,eta) &= U(x,t)e^beta \
t &= aeta
endalign

we would get the backward heat equation below

beginequation
U_t = - U_xx
endequation

However, the transversality condition becomes a problem, since as $eta rightarrow infty$, $e^beta rightarrow infty$.

Usually, if the terminal condition is of the form

beginequation
W(x,H) = g(x)
endequation

with $H$ finite, we could "reverse" it, that is, we could apply the following transformation:

beginequation
nu = H - eta
endequation

to obtain

beginequation
-W_nu + aW_xx - bW = 0
endequation

s.t.

beginequation
W(x,0) = g(x)
endequation

which we can solve the traditional way (Fourier transform). However, as my terminal condition happens only at infinity I can't apply the reverse transformation above, thus I don't know how to overcome this problem. Any hint or reference?

pde heat-equation

share|cite|improve this question

edited Apr 2 at 14:56

Nicolas Pimentel de Souza

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you tried solving the equation with $U(x,T)=g(x)$ and let $Ttoinfty$?
  $endgroup$
  – Dylan
  Apr 2 at 7:23
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I have edited the question, now it reflects properly the problem I have.
  $endgroup$
  – Nicolas Pimentel de Souza
  Apr 2 at 14:48
  

  
  
  
  

add a comment | 

$begingroup$

I have a backward parabolic equation of the form:

beginequation
W_eta + aW_xx - bW = 0
endequation

s.t.

beginequation
lim_eta rightarrow infty(x,eta) = g(x)
endequation

were $x in mathbbR$, $eta geqslant 0$, and $a,b$ are positive constants.

Applying the following transformations:

beginalign
W(x,eta) &= U(x,t)e^beta \
t &= aeta
endalign

we would get the backward heat equation below

beginequation
U_t = - U_xx
endequation

However, the transversality condition becomes a problem, since as $eta rightarrow infty$, $e^beta rightarrow infty$.

Usually, if the terminal condition is of the form

beginequation
W(x,H) = g(x)
endequation

with $H$ finite, we could "reverse" it, that is, we could apply the following transformation:

beginequation
nu = H - eta
endequation

to obtain

beginequation
-W_nu + aW_xx - bW = 0
endequation

s.t.

beginequation
W(x,0) = g(x)
endequation

which we can solve the traditional way (Fourier transform). However, as my terminal condition happens only at infinity I can't apply the reverse transformation above, thus I don't know how to overcome this problem. Any hint or reference?

pde heat-equation

share|cite|improve this question

edited Apr 2 at 14:56

Nicolas Pimentel de Souza

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

$endgroup$

share|cite|improve this question

edited Apr 2 at 14:56

Nicolas Pimentel de Souza

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

share|cite|improve this question

edited Apr 2 at 14:56

Nicolas Pimentel de Souza

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236

asked Apr 1 at 18:06

Nicolas Pimentel de SouzaNicolas Pimentel de Souza

236