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How do I obtain a solution of this Helmholtz equation?

0

$begingroup$

I want to get a solution of the following Helmholtz equation.

$$ Delta u+ k^2 u = f, qquad x in mathbbR^n, ; k>0$$

Using the Fourier transform, I have

beginalign*u&=int_mathbbR^n e^ixcdotxi frac1k^2- int_mathbbR^n e^-iycdotxi f(y)dydxi \
&=int_mathbbR^nf(y)int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi dy.
endalign*

But I can't calculate

$$int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi$$

How do I solve it?

If $n=3$, I probably know the answer that it might be $frace^ik$ although I don't know how it is derived.

integration pde harmonic-analysis linear-pde

share|cite|improve this question

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

$endgroup$

add a comment | 

0

$begingroup$

I want to get a solution of the following Helmholtz equation.

$$ Delta u+ k^2 u = f, qquad x in mathbbR^n, ; k>0$$

Using the Fourier transform, I have

beginalign*u&=int_mathbbR^n e^ixcdotxi frac1k^2- int_mathbbR^n e^-iycdotxi f(y)dydxi \
&=int_mathbbR^nf(y)int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi dy.
endalign*

But I can't calculate

$$int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi$$

How do I solve it?

If $n=3$, I probably know the answer that it might be $frace^ik$ although I don't know how it is derived.

integration pde harmonic-analysis linear-pde

share|cite|improve this question

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

$endgroup$

add a comment | 

$begingroup$

I want to get a solution of the following Helmholtz equation.

$$ Delta u+ k^2 u = f, qquad x in mathbbR^n, ; k>0$$

Using the Fourier transform, I have

beginalign*u&=int_mathbbR^n e^ixcdotxi frac1k^2- int_mathbbR^n e^-iycdotxi f(y)dydxi \
&=int_mathbbR^nf(y)int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi dy.
endalign*

But I can't calculate

$$int_mathbbR^n e^i(x-y)cdotxi frac1k^2 - dxi$$

How do I solve it?

If $n=3$, I probably know the answer that it might be $frace^ik$ although I don't know how it is derived.

integration pde harmonic-analysis linear-pde

share|cite|improve this question

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

$endgroup$

share|cite|improve this question

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

share|cite|improve this question

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

edited Mar 24 at 6:51

Rodrigo de Azevedo

13.2k41962

asked Mar 24 at 6:31

IdkwhatIdkwhat

507

asked Mar 24 at 6:31

IdkwhatIdkwhat

507