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Strongly singular integral
Posted 5 oct. 2010, 15:41 UTC−4 2 Replies
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Hi everybody,
I have a problem with a singularity in my equation.
I try to solve the integral (from physic book) : int (for x = 1 to x = 3) [ Ko (x-2) *( x +1)] dx
-This 1D-integral is performed for x = 2 to x = 3;
-Ko (x) is the 0-modified Bessel's function of second kind.
The difficulty is that Ko(x - 2) is divergent for x = 2. Thus the book explains that we have to perform this integral by removing the point x = 2 (it's like an integral defined using Cauchy principal value for strongly singular integral).
How perform my integral whose the kernel diverges in x = 2 ?
*******************************************************************************************
In my side, I use (I made) the function "nonzero" : Function Name : nonzero
Argument : x
Expression : ((x)+((x)==0))
I mean, I performed the integral : int (for x = 1 to x = 3) [ Ko (nonzero(x-2)) *( x +1)] dx using the application Weak Form Boundary.
*********************************************************************************************
What do you thing of this method ? Have you other idea to perform this integral removing x = 2 ?
More generally, can COMSOL performed Cauchy principal value for strongly singular integral ?
The COMSOL's books do not mention this type of problems (integrals with kernel strongly singular).
Thank you for your help.
Antony Sullivan
I have a problem with a singularity in my equation.
I try to solve the integral (from physic book) : int (for x = 1 to x = 3) [ Ko (x-2) *( x +1)] dx
-This 1D-integral is performed for x = 2 to x = 3;
-Ko (x) is the 0-modified Bessel's function of second kind.
The difficulty is that Ko(x - 2) is divergent for x = 2. Thus the book explains that we have to perform this integral by removing the point x = 2 (it's like an integral defined using Cauchy principal value for strongly singular integral).
How perform my integral whose the kernel diverges in x = 2 ?
*******************************************************************************************
In my side, I use (I made) the function "nonzero" : Function Name : nonzero
Argument : x
Expression : ((x)+((x)==0))
I mean, I performed the integral : int (for x = 1 to x = 3) [ Ko (nonzero(x-2)) *( x +1)] dx using the application Weak Form Boundary.
*********************************************************************************************
What do you thing of this method ? Have you other idea to perform this integral removing x = 2 ?
More generally, can COMSOL performed Cauchy principal value for strongly singular integral ?
The COMSOL's books do not mention this type of problems (integrals with kernel strongly singular).
Thank you for your help.
Antony Sullivan
2 Replies Last Post 6 oct. 2010, 05:31 UTC−4
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Posted:
1 decade ago
Hi
you could try to replace "0" by "eps" (2.22E-16 on my PC, it's the smallest representative number in the binary system we have on our FPU i.e. instead of dividing by "0" you use a if(T==0,eps,T) or simething along that order.
It could work, with the risk of generating a NAN
--
Good luck
Ivar
you could try to replace "0" by "eps" (2.22E-16 on my PC, it's the smallest representative number in the binary system we have on our FPU i.e. instead of dividing by "0" you use a if(T==0,eps,T) or simething along that order.
It could work, with the risk of generating a NAN
--
Good luck
Ivar
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Posted:
1 decade ago
Hi Ivar,
Thank you for your answer.
In fact, I would like to perform this integral (noted I) for x = 1 to x = 3, with I = 0 en x = 2 (in other words, I impose that I = 0 at the singularity x = 2 to avoid the divergence). Unfortunately, there is no predefined function in comsol that allows me to calculate this integral well defined...I don't see a general method to solve it.
Ivar, have you a "trick" to impose I = 0 at x = 2 and perform the rest of the integral ?
Thank you for your patience.
Antony Sullivan
Thank you for your answer.
In fact, I would like to perform this integral (noted I) for x = 1 to x = 3, with I = 0 en x = 2 (in other words, I impose that I = 0 at the singularity x = 2 to avoid the divergence). Unfortunately, there is no predefined function in comsol that allows me to calculate this integral well defined...I don't see a general method to solve it.
Ivar, have you a "trick" to impose I = 0 at x = 2 and perform the rest of the integral ?
Thank you for your patience.
Antony Sullivan