Need help finding solution to Bessel differential equation

[Fr4ct4l]

Hello all, need help with the following

I am deriving an analytical solution for a problem in petroleum engineering. It concerns fluid flow in porous media. Anyway, the equation is (see attachment)

P is pressure, it is a function of space in x-direction, so P(x)

d, THETA, and z are just constants

What is the general solution ?

I think I have to 'transform' it to a version amenable to the 'modified bessel function equation', but I am not sure how to do this.

I have the boundary conditions, I can take it from there...

Any help would be greatly appreciated ! I am doing this for my masters thesis, this is not just a classic PDE class exercise.

 

[Sourabh N]

What about y?

 

[Fr4ct4l]

Thank you Sourabh, made a mistake there, that's supposed to be an x ... please find corrected equation attached.

 

[Sourabh N]

Do we have any constraints on theta? Real or complex? iF real, can it take only integral values or anything?
Knowing this will help in finding the analyticity of the coefficient and we can use the standard functions (e.g bessel).
Also, are you looking for solutions for particularly small (or large) values of x? If yes, this will help us simplify the differential equation.

 

[Fr4ct4l]

d and theta are positive reals , not necessarily integers, small numbers , no bigger than 5

z is a positive real whose range is probably 10^-3 to 10^3

I don't think it needs to become so complicated as to need integrals or complex numbers. I am an engineer, not a mathematician , so as you can imagine if i present a solution with complex numbers and integrals to my fellow (less mathematically-inclined) peers, they will scream and burn me alive. They want nice simple things you can apply on the field with a pocket calculator (exaggerating ^^).

x will actually be plotted in log scale , so its range is probably from 10^-3 to 10^3.
I don't think for the purposes of my research we can jump straight into an asymptotic solution, if that's what you had in mind.

Idea : variable substitution ?

 

[Sourabh N]

I am sorry I don't have any good answer for you. You could try a power series solution?

I am unable to see a good variable substitution for this equation. Do you have anything specific in mind?

 

[Fr4ct4l]

Thank you for trying, I will keep trying (I have no choice hehe) and let you know if I come up with something.