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Solutions of linear PDE in the sense of distributions

1

$begingroup$

Let us consider the wave equation written in the form
$$
partial_ttphi-Deltaphi+phi=0.
$$
By a Fourier transform, this takes the form
$$
(k_0^2-bf k^2-1)phi(k)=0.
$$
This has as a solution, in the sense of distributions,
$$
phi(k)=A(k)delta(k_0^2-bf k^2-1).
$$
It is easy to see that, by Fourier transforming back,
$$
phi(x)=intfracd^3k(2pi)^3left(fracA(bf k,sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x-isqrtbf k^2+1t-fracA(bf k,-sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x+isqrtbf k^2+1tright)
$$
that appears as a general solution of the equation we started from.

My questions are the following:

• Is this approach sound to solve a linear PDE?


• What does it happen to the Dirac equation $(igammacdotpartial-1)psi=0$ using this technique?

pde distribution-theory

share|cite|improve this question

edited Mar 31 at 14:11

Jon

asked Mar 27 at 15:35

JonJon

4,43211223

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  As long as $phi$ can be assumed to not grow too fast (so that it is a Fourier transformable distribution), it is sound to use the Fourier transform to solve a linear PDE.
  $endgroup$
  – md2perpe
  Mar 30 at 23:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot @md2perpe. Indeed, it appears that the textbook solution comes out naturally, provided I change the integration variable as $bf krightarrow -bf k$ in the second term.
  $endgroup$
  – Jon
  Mar 30 at 23:09
  

  
  
  
  

add a comment | 

1

$begingroup$

Let us consider the wave equation written in the form
$$
partial_ttphi-Deltaphi+phi=0.
$$
By a Fourier transform, this takes the form
$$
(k_0^2-bf k^2-1)phi(k)=0.
$$
This has as a solution, in the sense of distributions,
$$
phi(k)=A(k)delta(k_0^2-bf k^2-1).
$$
It is easy to see that, by Fourier transforming back,
$$
phi(x)=intfracd^3k(2pi)^3left(fracA(bf k,sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x-isqrtbf k^2+1t-fracA(bf k,-sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x+isqrtbf k^2+1tright)
$$
that appears as a general solution of the equation we started from.

My questions are the following:

• Is this approach sound to solve a linear PDE?


• What does it happen to the Dirac equation $(igammacdotpartial-1)psi=0$ using this technique?

pde distribution-theory

share|cite|improve this question

edited Mar 31 at 14:11

Jon

asked Mar 27 at 15:35

JonJon

4,43211223

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  As long as $phi$ can be assumed to not grow too fast (so that it is a Fourier transformable distribution), it is sound to use the Fourier transform to solve a linear PDE.
  $endgroup$
  – md2perpe
  Mar 30 at 23:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks a lot @md2perpe. Indeed, it appears that the textbook solution comes out naturally, provided I change the integration variable as $bf krightarrow -bf k$ in the second term.
  $endgroup$
  – Jon
  Mar 30 at 23:09
  

  
  
  
  

add a comment | 

$begingroup$

Let us consider the wave equation written in the form
$$
partial_ttphi-Deltaphi+phi=0.
$$
By a Fourier transform, this takes the form
$$
(k_0^2-bf k^2-1)phi(k)=0.
$$
This has as a solution, in the sense of distributions,
$$
phi(k)=A(k)delta(k_0^2-bf k^2-1).
$$
It is easy to see that, by Fourier transforming back,
$$
phi(x)=intfracd^3k(2pi)^3left(fracA(bf k,sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x-isqrtbf k^2+1t-fracA(bf k,-sqrtbf k^2+1)2sqrtbf k^2+1e^ibf kcdotbf x+isqrtbf k^2+1tright)
$$
that appears as a general solution of the equation we started from.

My questions are the following:

• Is this approach sound to solve a linear PDE?


• What does it happen to the Dirac equation $(igammacdotpartial-1)psi=0$ using this technique?

pde distribution-theory

share|cite|improve this question

edited Mar 31 at 14:11

Jon

asked Mar 27 at 15:35

JonJon

4,43211223

$endgroup$

share|cite|improve this question

edited Mar 31 at 14:11

Jon

asked Mar 27 at 15:35

JonJon

4,43211223

share|cite|improve this question

edited Mar 31 at 14:11

Jon

asked Mar 27 at 15:35

JonJon

4,43211223

asked Mar 27 at 15:35

JonJon

4,43211223

asked Mar 27 at 15:35

JonJon

4,43211223