# Need help finding solution to Bessel differential equation

• Fr4ct4l
In summary, the equation is P(x) d, THETA, and z are 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 !The equation is a PDE, and it concerns fluid flow in porous media. The equation is y = P(x). The constants are d, THETA, and z. The general solution is unknown. The equation can be transformed to a version amenable to the 'modified bessel function 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.

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

#### Attachments

• equation.jpeg
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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.

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 ?

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?

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

## 1. What is a Bessel differential equation?

A Bessel differential equation is a type of ordinary differential equation that involves a Bessel function, which is a special mathematical function that is used to describe oscillatory phenomena in various fields of science and engineering.

## 2. What is the purpose of solving a Bessel differential equation?

The main purpose of solving a Bessel differential equation is to find a mathematical solution that accurately describes the physical behavior of a system or phenomenon in which Bessel functions are present. This can provide valuable insights and predictions for a wide range of real-world applications.

## 3. How do you solve a Bessel differential equation?

There are several methods for solving Bessel differential equations, including using power series, Frobenius method, and integral transforms. The specific approach used will depend on the form of the equation and the boundary conditions of the problem.

## 4. What are some common applications of Bessel differential equations?

Bessel differential equations can be found in many areas of science and engineering, including physics, astronomy, acoustics, electromagnetics, and signal processing. They are often used to model phenomena such as heat conduction, vibration of circular membranes, and electromagnetic radiation in cylindrical systems.

## 5. Are there any software programs available for solving Bessel differential equations?

Yes, there are various software programs and libraries that can help with solving Bessel differential equations, such as MATLAB, Mathematica, and SciPy. These programs often have built-in functions for Bessel functions and their derivatives, making it easier to solve complex equations.

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