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 2020-8-17

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编辑： hosein Javan 2020-8-19

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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hosein Javan 2020-8-18

David Goodmanson. hello again sir. I think this is all you need to know. the derivate of I_1 and K_1 can be rewritten in terms of I_0 and K_0.

you're right. I'll give a more thorough explanation. the PDE is an inhomogeneous laplace equation in 2d plane:

where R(x) is a periodic function at x with its main period equal to "L". and at y1<y<y2.

the solution is:

where:

"L" is the (minimum) length of periodicity of f(x,y) which occurs in x coordinate.

"n" is the harmonic number.

the "

" is a particular solution.

the "

" and "

" are integral coefficients and are unknown. these are also y-dependant but in each area they are constants. as following:

by applying boundary conditions we can find the above integral coefficients. (it is quite complicated to explain here but in one sentence it concerns the function and its derivative continuity at each y0 to y6).

now after this we have "n" systems of linear equations in the form:

the elements of A matrix are one of like these:

the goal of the program is to:

compute the integral coefficients matrix [C_n]
compute the solution using the general solution and the integral coefficients [C_n]

the problem appears in both steps when computing

and the rest (the second kind and the zero-order bessel functions). especially at high harmonic numbers of "n", and high values of "p".

-------------------------------

as for unit scaling, I have already tried that. normally both x and y coordinates are in term of meters. now if we change the unit of y, then the unit of L scales the same as well and the result does not change. therefore the input arguement of the bessel functions is dimensionless. unless it is in terms of radians and maybe we can replace by something? not sure.
there's another thing I'm thinking that I'm not sure. we wanted the accuracy of f(x,y) at first place and not its bessel functions. maybe the accuracy of bessel functions and their coefficients when they multiply, cancel each other and result into a good accuracy of "f".

-------------------------------

however rigth now I can not set a value lower than L=0.5 when trying to solve for n=99 harmonics.

I hope this time I've explained exactly. sorry again for too much writing.

hosein Javan 2020-8-19

编辑：hosein Javan 2020-8-19

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 2020-8-19

编辑：hosein Javan 2020-8-19

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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