How can I prove the following integral identity?
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The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $Delta t$ approaches zero. Consequently, I want to prove the following integral identity.
$$lim_{Delta t rightarrow 0}frac{1}{2 Delta t} int_{0}^infty x F(x) e^ {-frac{x^2 + r^2}{4 Delta t}} I_0(frac{x r}{2 Delta t}) {d}x = F(r)$$
where $F(r)$ is the initial solution of the problem whose solution is displayed by the LHS of the previous equation. $I_0$ is modified Bessel function of the first kind of zeroth order.
integration improper-integrals indefinite-integrals
asked Dec 23 '18 at 18:44
GalalGalal
344
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|
show 1 more comment
$begingroup$
The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $Delta t$ approaches zero. Consequently, I want to prove the following integral identity.
$$lim_{Delta t rightarrow 0}frac{1}{2 Delta t} int_{0}^infty x F(x) e^ {-frac{x^2 + r^2}{4 Delta t}} I_0(frac{x r}{2 Delta t}) {d}x = F(r)$$
where $F(r)$ is the initial solution of the problem whose solution is displayed by the LHS of the previous equation. $I_0$ is modified Bessel function of the first kind of zeroth order.
integration improper-integrals indefinite-integrals
asked Dec 23 '18 at 18:44
GalalGalal
344
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$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
1
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What is $I_0$? Is it a Bessel function?
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– rafa11111
Dec 23 '18 at 18:53
1
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01
|
show 1 more comment
$begingroup$
The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $Delta t$ approaches zero. Consequently, I want to prove the following integral identity.
$$lim_{Delta t rightarrow 0}frac{1}{2 Delta t} int_{0}^infty x F(x) e^ {-frac{x^2 + r^2}{4 Delta t}} I_0(frac{x r}{2 Delta t}) {d}x = F(r)$$
where $F(r)$ is the initial solution of the problem whose solution is displayed by the LHS of the previous equation. $I_0$ is modified Bessel function of the first kind of zeroth order.
integration improper-integrals indefinite-integrals
asked Dec 23 '18 at 18:44
GalalGalal
344
$endgroup$
The following equation is a solution of a convection-diffusion problem. I want to prove that the resultant solution is simplified to the initial condition when $Delta t$ approaches zero. Consequently, I want to prove the following integral identity.
$$lim_{Delta t rightarrow 0}frac{1}{2 Delta t} int_{0}^infty x F(x) e^ {-frac{x^2 + r^2}{4 Delta t}} I_0(frac{x r}{2 Delta t}) {d}x = F(r)$$
where $F(r)$ is the initial solution of the problem whose solution is displayed by the LHS of the previous equation. $I_0$ is modified Bessel function of the first kind of zeroth order.
integration improper-integrals indefinite-integrals
integration improper-integrals indefinite-integrals
asked Dec 23 '18 at 18:44
GalalGalal
344
asked Dec 23 '18 at 18:44
GalalGalal
344
edited Dec 23 '18 at 18:55
Galal
asked Dec 23 '18 at 18:44
GalalGalal
344
asked Dec 23 '18 at 18:44
GalalGalal
344
asked Dec 23 '18 at 18:44
GalalGalal
344
344
$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
1
$begingroup$
What is $I_0$? Is it a Bessel function?
$endgroup$
– rafa11111
Dec 23 '18 at 18:53
1
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01
|
show 1 more comment
$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
1
$begingroup$
What is $I_0$? Is it a Bessel function?
$endgroup$
– rafa11111
Dec 23 '18 at 18:53
1
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01
$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
1
1
$begingroup$
What is $I_0$? Is it a Bessel function?
$endgroup$
– rafa11111
Dec 23 '18 at 18:53
$begingroup$
What is $I_0$? Is it a Bessel function?
$endgroup$
– rafa11111
Dec 23 '18 at 18:53
1
1
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01
|
show 1 more comment
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$begingroup$
Thanks for your comment.
$endgroup$
– Galal
Dec 23 '18 at 18:51
1
$begingroup$
What is $I_0$? Is it a Bessel function?
$endgroup$
– rafa11111
Dec 23 '18 at 18:53
1
$begingroup$
It is the modified Bessel function of the first kind of zero order.
$endgroup$
– Galal
Dec 23 '18 at 18:55
$begingroup$
Thanks for editing the question to address the definition of $I_0$! I guess that the proof will have something to do with the Dirac delta 'function' and its 'sifting' property.
$endgroup$
– rafa11111
Dec 23 '18 at 18:58
$begingroup$
I am not sure I understand you. The equation doesn't include Dirac delta function and this solution has not been obtained using it.
$endgroup$
– Galal
Dec 23 '18 at 19:01