2 2 2 ( ) 0 xy xy x p y ′′ ′ + + - = () () p yx B x = ( 29 () () 0 d hxy xy a x b dx ψ ′ = < < ( ), () x h x ψ ′ () () () 0 Pxy Qxy Rxy ′′ ′ + + = () () () () () () 0 xPxy xQxy xRxy μ μ μ ′′ ′ + + = ( 29 () () () () d xPx xQx dx d Q P P P Q dx P μ μ μ μ μ μ μ = ′ - ′ ′ ⇒ + = ⇒ = () () A Q x Exp dx Px P μ   =     ∫ World Applied Sciences Journal 15 (12): 1687-1691, 2011 ISSN 1818-4952 © IDOSI Publications, 2011 Corresoponding Author: Delkhosh Mehdi, Department of Computer, Islamic Azad University, Bardaskan Branch, Bardaskan, Iran. Tel: +985327226200, E-mail: mehdidelkhosh@yahoo.com. 1687 The Conversion a Bessel’s Equation to a Self-Adjoint Equation and Applications Delkhosh Mehdi Department of Computer, Islamic Azad University, Bardaskan Branch, Bardaskan, Iran Abstract: In many applications of various Self-adjoint differential equations, whose solutions are complex, are produced [12, 15]. In this paper, a method for the conversion Bessel equation to Self-adjoint equation to provide a method and then, as the inverse of the transformation Self-adjoint equations of the Bessel equation. By which one can obtain analytical solutions to Self-adjoint equations. Because this solution, an exact analytical solution can provide to us, we benefited from the solution of numerical Self-adjoint equations [7, 9, 11, 13, 14, 16]. Key word: Bessel’s equation %Bessel’s function %Self-adjoint equation %Self-adjointization factor INTRODUCTION (1) In many applications of science to solve many differential equations, we find that these equations are Where h(x)>0 on (a , b) and are continuous Self-adjoint equations and solve relatively complex because they're forced to use numerical methods, which are contained several errors [7, 9, 11, 13, 14, 16]. There are several methods for solving equations, there one of which can be seen in the literature [4, 5, 6, 12, 15], where the change of variables is very complicated to use. In this paper, for solving analytical Self-adjoint equations, we get a method which the first, Self-adjoint equation into a Bessel equation and then using the solutions of Bessel equation, for an exact and analytical solution a Self-adjoint equation. C Before going to the main point, we start to introduce three following equations: Bessel’s Equation: The Bessel’s equation is as follows [1, 2, 3, 8 and 10]: Where P is a non-negative real number. To simplify, we present the general solution of Bessel’s Equation as follows: Self-Adjoint Equation: A second order linear homogeneous differential equation is called self-adjoint if And only if it has the following form [2, 12 and 15]: functions on [a,b]. Self-Adjointization Factor: By multiplying both sides a second order linear homogeneous equation in a function μ (x), it can be changed into a self-adjoint equation. Namely, we consider the following linear homogeneous equation: (2) Where P(x) is a non-zero function on [a,b]. By multiplying both sides in μ (x), we have If we check the self-adjoint condition, we have: Thus (3) Where A is a real number that will be specified exactly during the process. If we multiply both sides of (2) and (3) equations in each other, then we have the following form of self-adjoint equation: