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Explicit wave-package solution of the Klein Gordon equation

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0

$begingroup$

I need to generate initial for a 3D Klein Gordon (KG) solver. Therefore I'm interested in physical meaningful wave-package solutions. By considering the free KG equation
beginequation
Box psi(t,x) = psi(t,x), quad (x,t) in Omega times mathbbR^+,
endequation
an obvious wave-package solution is given by
beginequation
psi(x,t) = int e^-k^2 +i x cdot k - iomega(k)t dk,
endequation
with $omega(k)=sqrtk^2+1$. From this formula we immediatly see that
beginequation
psi(x,0) = e^-fracx^24 +i x cdot k,
endequation
and the time derivative at $t=0$ becomes
beginequation
partial_t psi(x,0) = -i int omega(k) e^-k^2 +i x cdot k dk.
endequation
And one can recognize this as the Fourier transform of
beginequation
mathcalFleft( omega(k) e^-k^2 right).
endequation
So far so good, the idea was just to calculate the FFT of the function $omega(k) e^-k^2$ on my array but that isn't so accurate as it can be by an explicit expression of the initial time derivative. So my question if anyone has an idea how to write the soultion explicitly. (this can also include the erf function as well)

integration numerical-methods fourier-analysis

share|cite|improve this question

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

$endgroup$

add a comment | 

0

$begingroup$

I need to generate initial for a 3D Klein Gordon (KG) solver. Therefore I'm interested in physical meaningful wave-package solutions. By considering the free KG equation
beginequation
Box psi(t,x) = psi(t,x), quad (x,t) in Omega times mathbbR^+,
endequation
an obvious wave-package solution is given by
beginequation
psi(x,t) = int e^-k^2 +i x cdot k - iomega(k)t dk,
endequation
with $omega(k)=sqrtk^2+1$. From this formula we immediatly see that
beginequation
psi(x,0) = e^-fracx^24 +i x cdot k,
endequation
and the time derivative at $t=0$ becomes
beginequation
partial_t psi(x,0) = -i int omega(k) e^-k^2 +i x cdot k dk.
endequation
And one can recognize this as the Fourier transform of
beginequation
mathcalFleft( omega(k) e^-k^2 right).
endequation
So far so good, the idea was just to calculate the FFT of the function $omega(k) e^-k^2$ on my array but that isn't so accurate as it can be by an explicit expression of the initial time derivative. So my question if anyone has an idea how to write the soultion explicitly. (this can also include the erf function as well)

integration numerical-methods fourier-analysis

share|cite|improve this question

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

$endgroup$

add a comment | 

$begingroup$

I need to generate initial for a 3D Klein Gordon (KG) solver. Therefore I'm interested in physical meaningful wave-package solutions. By considering the free KG equation
beginequation
Box psi(t,x) = psi(t,x), quad (x,t) in Omega times mathbbR^+,
endequation
an obvious wave-package solution is given by
beginequation
psi(x,t) = int e^-k^2 +i x cdot k - iomega(k)t dk,
endequation
with $omega(k)=sqrtk^2+1$. From this formula we immediatly see that
beginequation
psi(x,0) = e^-fracx^24 +i x cdot k,
endequation
and the time derivative at $t=0$ becomes
beginequation
partial_t psi(x,0) = -i int omega(k) e^-k^2 +i x cdot k dk.
endequation
And one can recognize this as the Fourier transform of
beginequation
mathcalFleft( omega(k) e^-k^2 right).
endequation
So far so good, the idea was just to calculate the FFT of the function $omega(k) e^-k^2$ on my array but that isn't so accurate as it can be by an explicit expression of the initial time derivative. So my question if anyone has an idea how to write the soultion explicitly. (this can also include the erf function as well)

integration numerical-methods fourier-analysis

share|cite|improve this question

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

$endgroup$

share|cite|improve this question

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

share|cite|improve this question

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

edited Mar 22 at 0:03

J. W. Tanner

4,4891320

asked Mar 21 at 23:50

HamilcarHamilcar

404211

asked Mar 21 at 23:50

HamilcarHamilcar

404211