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my differential equation general solution form requires very high or low values, exceeding double precision capability, while I know its solution is finite

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hosein Javan
hosein Javan on 17 Aug 2020
Edited: hosein Javan on 19 Aug 2020
as a brief explanation, I'm trying to solve a first order bessel equation which is part of a more complicated PDE problem, whose solution is as following form:
now we have a bunch of these functions and not one, so there are for example 10 integral coefficients like c1(1), c1(2),... and c2(1),c2(2),....
now when I apply the boundary conditions, a set of linear equations are made by which I can calculate c1, c2,... . in the form "A*C=B".
now as x is near zero, I1(x) goes to zero and K1(x) goes to inf, and as x goes up, I1(x) goes to inf and K1(x) goes to zero. given that y(x) is finite and something less than 0.07, c1 and c2 go to inf or zero depending on their factor. and here the problem begins. working with very high and low values decrease accuracy and might lead to inf or zero.
any idea to overcome this issue?

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David Goodmanson
David Goodmanson on 19 Aug 2020
Hello sir,
I don't have the impression you have written 'too much' and in fact I have a couple more questiions.
Could you mention what the physical situation is?
Something must distinguish the regions. It seems that each region has its own value of p determined in some fashion, is that correct?
For boundary conditions, is it as simple as for example
C4n I() + c5n K() = C6n I() + c7n K()
C4n dI()/dy + c5n dK()/dy = const* [ C6n dI()/dy + c7n dK()/dy ]
or
C4n d(yI())/dy + c5n d(yK())/dy = const* [ C6n d(yI())/dy + c7n d(yK())/dy ]
hosein Javan
hosein Javan on 19 Aug 2020
Hi Mr.Lee. thanks for having time on replying. unfortunately in my case, the product of I1(x)*K1(x) is of no use to me. but I was thinking at something else, what do you think if we can change the unit of radian to a larger unit so that the input of bessel function reduces. if I replace 2*pi by "k", what happens to my equations?
hosein Javan
hosein Javan on 19 Aug 2020
Hello again Mr. Goodmanson. thanks for your patience. yes the boundary condition is somehow like that. in fact it states that and must be continuous at each y1,...,y5. where "k" and "m" are constants for each "n" and has been defined before. and the regions are distinguished by y1,...,y5 boundaries. p is the frequency across only "x" and not "y", so for each region it is the same.
Mr. Goodmanson what about this idea: the input of bessel function dimension is radian which is "dimensionless", however I was wondering if we change that unit, what change should I do to the equations?

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