The Application of Bessel Function in the Definite Solution Problem of Cylindrical Coordinate System

The variable separation method is an important method to solve the definite solution problems, especially the definite solution problems of cylinder and sphere regions. This method can solve these problems on cylinder and sphere regions, but the solving procedures are very difficult in the practical application. It is often solved by combining the properties of Bessel functions. In this paper, we propose a method combining Bessel function to solve homogeneous definite solution problem on the cylindrical coordinate system and give the algorithm of solving a definite problem. This algorithm is easy to implement and simplifies the process of calculation. Firstly, the definition and properties of Bessel function are briefly recalled, which are the first and essential step to solve the definite solution problem. Then we give the basic process of solving homogeneous definite solution problem, where consider the problem of the definite solution of the homogeneous wave equation, homogeneous heat conduction equation and Laplace equation. We analyze the solution of the Bessel equation definite solution problem under three kinds of boundary conditions and conclude the algorithm of solving a definite problem. At last, two numerical examples are provided to validate the feasibility of the proposed method.


Introduction
Bessel function is one of the most significant special functions, which is widely used in atmospheric science, mechanics, mathematics and other disciplines. Bessel function is obtained when equation Helmholtz and Laplace equation are solved by separating variables in cylindrical or spherical coordinates [12]. There are some limitations to use elementary function to solve the definite problem. Therefore, Bessel functions have been attracted considerable attention. The solution of the definite solution problem usually needs to be converted into the partial differential equation with variable coefficient in cylindrical or spherical coordinate system, then solve by using special functions. There are many classical problems, such as electromagnetic wave propagation waveguides [4,7], heat conduction problem [1], the vibration mode problem of circular or annular films and so on.
Several special cases of positive integer order of Bessel function were proposed by Swiss mathematician Daniel Bernoulli as early as the mid-18th century. Bessel functions first appeared in problems involving catenary oscillation, cooling of long cylinders and tension membrane vibration [6]. In 1824, German mathematician F. W. Bessel systematically put forward the overall theoretical framework of Bessel function for the first time. Rossetti [2] deduced an approximate form for the standard Bessel functions of first and second kind and obtained the real zeros. The definition and properties of Bessel function were introduced in detail in [13]. Karatsuba [3] presented a fast method to calculate Bessel function. Then fast and accurate Bessel function computations were presented in [5]. Zhou [9] applied the deformed Bessel function to mechanics and obtained the formal solution in mechanical analysis. Recently, many scholars were concerned about the application of Bessel functions for solving equations [15,16]. Bessel function was applied to the definite solution of heat conduction equation [8]. The mixed problems of axisymmetric parabolic partial differential equations and spherical symmetric parabolic equations in cylindrical regions are solved by using Bessel function [11].
In this paper, we propose a method combining Bessel Cylindrical Coordinate System function to solve homogeneous definite solution problem of cylindrical coordinate system. The definite solution problems with different boundary conditions are analyzed. A brief outline of this paper is as follows. Section 2 recalls the definition and properties of the Bessel function, which provides a theoretical basis for the following methods. In Section 3, the method of separating variables to solve the homogeneous definite solution problem is summarized, and Bessel equation definite solution problem under three kinds of boundary conditions are analyzed. Numerical examples are provided to validate the proposed method in Section 4. Finally, the paper is concluded.

Bessel Function
Bessel function is the solution of Bessel equation, except elementary function. Bessel function is the most commonly used function in mathematics, physics and engineering.

Bessel Function Definition
Bessel's equation is the equation where v is a constant and is called order of equation, which can be any real number or complex number. The solutions of Bessel's equation are called as Bessel functions, which can be divided into three kinds. The first kind of Bessel functions are often called Bessel functions, which are denoted by ( ) The second kind of Bessel function are often called Neumann function, which is denoted by ( ) v Y x . This can be written in the form The third kind of Bessel functions often are called Henkel functions, which are denoted by (1) x . These can be written in the form x are respectively called the first Henkel function and the second Henkel function.

Properties of Bessel Function
Bessel functions are characterized by many important properties. Analyzing its properties was the primary step in understanding the Bessel function.

Recursion Formula
Bessel functions have recursive relationships. Taking the first kind of Bessel functions as an example, we have the following recursive formulas .
Similar properties exist in other kind of functions. is close to π .

Zero Point
We know that the zero point of , then we can obtain the eigenvalues and eigenfunctions, , .

Bessel Function Series Expansion
Arbitrary function ( ) f x has a continuous first order derivative and the second derivative piecewise continuous in

Solving the Definite Solution Problem
The separation of variables method is the most common and important method to solve the problem of definite solution of partial differential equations, which is widely used in various definite solution problems. Bessel functions make it easier to solve the definite solution problem in cylindrical coordinate system. We consider the problem of the definite solution of the homogeneous wave equation, homogeneous heat conduction equation and Laplace equation.

Separating Variables
A set of ordinary differential equations is obtained by separating variables from the original equation.

Laplace Equation
The expression in cylindrical coordinate system is Let the form of variable separation be Substituting the above equation into (10), we obtain Multiplying both sides of the equation by It decomposes into two equations Dividing in both sides of (15) by 2 r and transfers, we get Eventually we can decompose the original equation into

Homogeneous Wave Equation
First, we separate time variables from space variables. Supposing that ( , ) It decomposes into two equations 2 2 '' 0,

Homogeneous Heat Conduction Equation
Similarly, first we separate the time variables, and then we separate the space variables in cylindrical coordinates. We can separate the original equation to get

General Condition
We consider the general case In the situation of ω<0, the general solution of the equation is ( ) .
In the situation of ω=0, the general solution of the equation is ( ) .
In the situation of ω>0, the general solution of the equation is Then the solution sequence ( ), 1, 2, n X x n = ⋯ is obtained according to the boundary conditions.

Eigenvalue Problems of Bessel Equation
Supposing that x r λ = the equation can be reduced to Finally, from the third kind of boundary conditions According to the properties of Bessel function, the above equation can be rewritten as x means the n-th zero of (35).

Superposing Solution Sequences
From the above process, the formal solution can be obtained according to the principle of superposition ( , , , ) ( , , , ) Then we confirm the correlation coefficient according to the initial conditions. Finally, we solved the definite solution problem.
Consequently, we give an algorithm of solving a definite problem as following.

Algorithm Solving a definite problem
Step i Determine the type of equation; Step ii Obtain ordinary differential equations by separating variables; Step iii Obtain eigenvalues and eigenfunctions according to the boundary conditions; Step iv Solve the corresponding solution sequence; Step v Superposition all solutions; Step vi Confirm the correlation coefficient according to the initial conditions.

Numerical Example
In this section, we present numerical examples to illustrate our methods in the above sections.
Example 1 Consider the axisymmetric free vibration problem of an infinitely long cylinder with radius 0 0.5 r = . Suppose that the vibrational displacement of the particle is ( , , , ) u r z t θ . Clearly, the solution is independent of the variables , z θ . Thus, we separate the variables by introducing We can separate the original equation to get The solution of (38) is

Conclusion
The method of separating variables is usually used to solve the problem of definite solution of partial differential equations. We propose a method combining Bessel function to solve homogeneous definite solution problem of cylindrical coordinate system. Numerical examples validate the feasibility of the method. In addition, the cylindrical coordinate system introduced in this paper, Bessel function can also be used for the definite solution problem in the spherical coordinate system.