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System of integral equations with parameter not fixed
Posted 2010/06/04 5:06 GMT-4 3 Replies
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Hello everybody,
I use COMSOL for several months but unfortunately the problem I currently can not seem to be solved with COMSOL.
I have to solve a system of integral equations. There are two integral equations (these are two area integrals) and two unknown who are both functions of space (and I call u (x, y , z) and v (x, y, z)). This type of problem is easily solved by using COMSOL mode "General Form" (see attachment) OK ! But my problem is somewhat more complicated because of an additional term which I have not fixed ...
In my equations, so there is this additional term not fixed (I call LAMBDA and is independent of x, y, z) which appears nontrivially in these equations [LAMBDA term appears in the exponential type exp (-LAMBDA ^ 2 * ((x-x ') ^ 2 + (y-y') ^ 2 + (z-z ') ^ 2))].
According to old theoretical results about these equations, the system admits only discrete solutions for u (x, y, z) and v (x, y, z) (I call therefore u_i (x, y, z) and v_i (x, y, z)) and each of these solutions is associated with a value of LAMBDA (I call therefore Lambda_i) and also to be found. This system of integral equations seems somewhat analogous to an eigenvalue equation, but his writing does not appear explicitly as any other writing or elsewhere!
How COMSOL can solve this problem and find all solutions of this system (I mean find all u_i (x, y, z), v_i (x, y, z) and lambda_ i possible) ?
The solver used to solve an eigenvalue problem only works if my problem is written as A*u = LAMBDA * u... this is not the case here. On the other hand the solver "Stationary" only works if there is no term in LAMBDA. The "parametric solver'' is not very useful because LAMBDA not be fixed (even by varying).
Someone does a track to solve this problem with COMSOL? In fact I do not know if this can be treated by FEM...
Thank you for your help ! I am at your disposal for any further information on this model !
Sebastian
I use COMSOL for several months but unfortunately the problem I currently can not seem to be solved with COMSOL.
I have to solve a system of integral equations. There are two integral equations (these are two area integrals) and two unknown who are both functions of space (and I call u (x, y , z) and v (x, y, z)). This type of problem is easily solved by using COMSOL mode "General Form" (see attachment) OK ! But my problem is somewhat more complicated because of an additional term which I have not fixed ...
In my equations, so there is this additional term not fixed (I call LAMBDA and is independent of x, y, z) which appears nontrivially in these equations [LAMBDA term appears in the exponential type exp (-LAMBDA ^ 2 * ((x-x ') ^ 2 + (y-y') ^ 2 + (z-z ') ^ 2))].
According to old theoretical results about these equations, the system admits only discrete solutions for u (x, y, z) and v (x, y, z) (I call therefore u_i (x, y, z) and v_i (x, y, z)) and each of these solutions is associated with a value of LAMBDA (I call therefore Lambda_i) and also to be found. This system of integral equations seems somewhat analogous to an eigenvalue equation, but his writing does not appear explicitly as any other writing or elsewhere!
How COMSOL can solve this problem and find all solutions of this system (I mean find all u_i (x, y, z), v_i (x, y, z) and lambda_ i possible) ?
The solver used to solve an eigenvalue problem only works if my problem is written as A*u = LAMBDA * u... this is not the case here. On the other hand the solver "Stationary" only works if there is no term in LAMBDA. The "parametric solver'' is not very useful because LAMBDA not be fixed (even by varying).
Someone does a track to solve this problem with COMSOL? In fact I do not know if this can be treated by FEM...
Thank you for your help ! I am at your disposal for any further information on this model !
Sebastian
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3 Replies Last Post 2010/06/07 5:53 GMT-4
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Posted:
1 decade ago
Hi
I do not understand fully the definition of your "vint" and "uint", as these are not symmetric for the full sphere, normal ?
2nd what about "lambda" if its an unknown it requries some global equations so COMSOL would solve it, no ?
By the way, perhaps use another name or an uppercase "Lambda".
As "lambda" is often used internally in COMSOL and its easy to make duplicated names like that with surely funny results, (even if I do not believe this is the case for your non-eignenmode case)
Have fun Comsoling
Ivar
I do not understand fully the definition of your "vint" and "uint", as these are not symmetric for the full sphere, normal ?
2nd what about "lambda" if its an unknown it requries some global equations so COMSOL would solve it, no ?
By the way, perhaps use another name or an uppercase "Lambda".
As "lambda" is often used internally in COMSOL and its easy to make duplicated names like that with surely funny results, (even if I do not believe this is the case for your non-eignenmode case)
Have fun Comsoling
Ivar
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Posted:
1 decade ago
Hi Ivar,
In my model "vint" and "uint" are integrales defined on the full sphere. I corrected my previous model. This time the equations are integrated over the full sphere. Physics---->Subdomain Settings gives the full equations. The "stationary" solve is used by default.
In this problem, I try to solve a system of two integral equations (u(x,y,z) and v(x,y,z) are my unknown), but which include a parameter "Lambda" (not fixed and not defined by a third equation). This system admits some solutions only for some values of Lambda...I would like use COMSOL to :
1) find all values of lambda for which this system has solutions (noted Lambda_1, Lambda_2, ...)
2) then find these solutions ((u_1, v_1), (u_2, v_2)...).
This problem is however very easy to solve by hand (for simple geometries) and only two equations are enough !
Thank you for your help !
Sébastien
In my model "vint" and "uint" are integrales defined on the full sphere. I corrected my previous model. This time the equations are integrated over the full sphere. Physics---->Subdomain Settings gives the full equations. The "stationary" solve is used by default.
In this problem, I try to solve a system of two integral equations (u(x,y,z) and v(x,y,z) are my unknown), but which include a parameter "Lambda" (not fixed and not defined by a third equation). This system admits some solutions only for some values of Lambda...I would like use COMSOL to :
1) find all values of lambda for which this system has solutions (noted Lambda_1, Lambda_2, ...)
2) then find these solutions ((u_1, v_1), (u_2, v_2)...).
This problem is however very easy to solve by hand (for simple geometries) and only two equations are enough !
Thank you for your help !
Sébastien
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Posted:
1 decade ago
Hi
Well for me the requirement remain that you must tells COMSOL that Lambda is a variable for which it needs to find a solution, which could also be done by adding some "equations" since you say you know that Lambda is constant
I'll try to look at your model by tomorrow
ivar
Well for me the requirement remain that you must tells COMSOL that Lambda is a variable for which it needs to find a solution, which could also be done by adding some "equations" since you say you know that Lambda is constant
I'll try to look at your model by tomorrow
ivar