Analytic Solutions of Some Self-Adjoint Equations by Using Variable Change Method and Its Applications

Abstract

Many applications of various self-adjoint differential equations, whose solutions are complex, are produced (Arfken, 1985; Gandarias, 2011; and Delkhosh, 2011). In this work we propose a method for the solving some self-adjoint equations with variable change in problem, and then we obtain a analytical solutions. Because this solution, an exact analytical solution can be provided to us, we benefited from the solution of numerical Self-adjoint equations (Mohynl-Din, 2009; Allame and Azal, 2011; Borhanifar et al. 2011; Sweilam and Nagy, 2011; Gülsu et al. 2011; Mohyud-Din et al. 2010; and Li et al. 1996).

1. Introduction

Many applications of science to solve many differential equations, we find that these equations are self-adjoint equations and solve relatively complex because they are forced to use numerical methods, which are contained several errors [1–6].

There are several methods for solving equations, there one of which can be seen in the literature [7–11], where the change of variables is very complicated to use.

In this paper, for solving analytical some self-adjoint equations, we get a method with variable change in problem, and then we obtain a analytical solutions.

Before going to the main point, we start to introduce three following equations.

1.1. Self-Adjoint Equation

A second-order linear homogeneous differential equation is called self-adjoint if and only if it has the following form [10–13]: 𝑑ℎ(𝑥)𝑦𝑑𝑥^′+𝜓(𝑥)𝑦=0𝑎<𝑥<𝑏, (1.1) where ℎ(𝑥)>0 on (𝑎,𝑏) and 𝜓(𝑥), ℎ^′(𝑥) are continuous functions on [𝑎,𝑏].

1.2. Self-Adjointization Factor

By multiplying both sides, a second order linear homogeneous equation in a function 𝜇(𝑥) can be changed into a self-adjoint equation. Namely, we consider the following linear homogeneous equation: 𝑃(𝑥)𝑦^′′+𝑄(𝑥)𝑦^′+𝑅(𝑥)𝑦=0, (1.2) where𝑃(𝑥) is a non-zero function on [𝑎,𝑏].

By multiplying both sides in 𝜇(𝑥), we have 𝜇(𝑥)𝑃(𝑥)𝑦^′′+𝜇(𝑥)𝑄(𝑥)𝑦^′+𝜇(𝑥)𝑅(𝑥)𝑦=0. (1.3) If we check the self-adjoint condition, we have: 𝑑𝜇(𝑥)𝑃(𝑥)(𝑥)𝑄(𝑥)𝑑𝑥=𝜇⟹𝜇^′𝑃+𝜇𝑃^′=𝜇𝑄⟹𝑑𝜇𝜇=𝑄−𝑃^′𝑃𝑑𝑥. (1.4) Thus𝜇(𝑥)=𝐴𝑃(𝑥)𝑄Exp𝑃𝑑𝑥, (1.5) where 𝐴 is a real number that will be specified exactly during the process.

If we multiply both sides of (1.2) and (1.5) by each other, then we have the following form of self-adjoint equation:𝑑𝜇(𝑥)𝑃(𝑥)𝑦𝑑𝑥^′+𝜇(𝑥)𝑅(𝑥)𝑦=0. (1.6)

From now on, we will focus on the self-adjoint equations shown in (1.1).

1.3. Wronskian

The Wronskian of two functions 𝑓 and 𝑔 is [12, 14] 𝑊(𝑥)=𝑊𝑓,𝑔=𝑓^′𝑔−𝑓𝑔^′. (1.7) More generally, for 𝑛 real- or complex-valued functions𝑓₁,𝑓₂,…,𝑓_𝑛, which are 𝑛−1 times differentiable on an interval 𝐼, the Wronskian 𝑊(𝑥)=𝑊(𝑓₁,…,𝑓_𝑛) as a function on I is defined by 𝑊(𝑥)=𝑓₁⋯𝑓_𝑛𝑓⋮⋱⋮_1(𝑛−1)⋯𝑓_𝑛^(𝑛−1). (1.8) That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the (𝑛−1)st derivative, thus forming a square matrix sometimes called a fundamental matrix.

When the functions 𝑓_𝑖 are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel’s identity, even if the functions 𝑓_𝑖 are not known explicitly.

Theorem 1.1. If 𝑃(𝑥)𝑦^′′+𝑄(𝑥)𝑦^′+𝑅(𝑥)𝑦=0, then 𝑊(𝑥)=𝑒^{−∫(𝑄/𝑃)𝑑𝑥}. (1.9)

Proof. Let two solution of equation by 𝑦₁ and 𝑦₂, then, since these solutions satisfy the equation, we have 𝑃𝑦₁^′′+𝑄𝑦^′₁+𝑅𝑦₁=0,𝑃𝑦₂^′′+𝑄𝑦^′₂+𝑅𝑦₂=0. (1.10) Multiplying the first equation by 𝑦₂, the second by 𝑦₁, and subtracting, we find 𝑃⋅𝑦₁𝑦₂^′′−𝑦₂𝑦₁^′′+𝑄⋅𝑦₁𝑦^′₂−𝑦₂𝑦^′₁=0. (1.11) Since Wronskian is given by 𝑊=𝑦₁𝑦^′₂−𝑦₂𝑦^′, thus 𝑃⋅𝑑𝑊𝑑𝑥+𝑄⋅𝑊=0. (1.12) Solving, we obtain an important relation known as Abel’s identity, given by 𝑊(𝑥)=𝑒^{−∫(𝑄/𝑃)𝑑𝑥}. (1.13)

2. The Solving Some Self-Adjoint Equation

Now, we show that self-adjoint equation (1.1) is changeable to two linear differential equations: 𝑑ℎ(𝑥)𝑦𝑑𝑥^′(𝑥)(𝑥)𝑦+𝜓𝑦=0⟹ℎ^′′+ℎ^′(𝑥)𝑦^′+𝜓(𝑥)𝑦=0⟹𝑦^′′+ℎ^′(𝑥)ℎ(𝑥)𝑦^′+𝜓(𝑥)ℎ(𝑥)𝑦=0. (2.1) By replacing of change variable𝑦=𝑢(𝑥)⋅𝑣(𝑥), where 𝑢(𝑥) and 𝑣(𝑥) are continuous and differentiable functions, we obtain 𝑢^′′⋅𝑣+2𝑢^′⋅𝑣^′+𝑢⋅𝑣^′′+ℎ^′(𝑥)ℎ(𝑥)𝑢^′⋅𝑣+𝑢⋅𝑣^′+𝜓(𝑥)ℎ(𝑥)𝑢⋅𝑣=0, (2.2) or 𝑢^′′+2𝑣^′𝑣+ℎ^′ℎ𝑢^′+𝑣^′′+ℎ^′/ℎ𝑣^′+𝜓/ℎ⋅𝑣𝑣𝑢=0. (2.3) Now, 𝑢(𝑥) and 𝑣(𝑥) values are calculated with the following assumptions: 2𝑣^′𝑣+ℎ^′ℎ𝑣=0,_′′+ℎ^′ℎ𝑣^′+𝜓ℎ𝑣=0. (2.4)(2.5) Now, corresponding to equation (2.4), we have 𝑣(𝑥)=𝑒^{−1/2∫(ℎ′/ℎ)𝑑𝑥}=(ℎ(𝑥))^−1/2=1ℎ(𝑥)=𝑊(𝑥), (2.6) where 𝑊(𝑥) is Wronskian.

Also, corresponding to (2.4), (2.5) and (2.6), we have −ℎ^′′+2ℎ3ℎ^′24ℎ²ℎ𝑣+^′ℎ−ℎ^′𝜓2ℎ𝑣+ℎ𝑣=0, (2.7) or 𝜓(𝑥)=ℎ^′′2−ℎ^′24ℎ. (2.8) Thus, if in (1.1) the following relations are established 𝜓(𝑥)=ℎ^′′2−ℎ^′24ℎ, (2.9) then, we have from (2.3), (2.4), (2.5), and (2.6): 𝑣(𝑥)=1ℎ(𝑥),𝑢(𝑥)=𝐶₁𝑥+𝐶₂, (2.10) and, the answer to self-adjoint equation (1.1) will be 𝑦(𝑥)(𝑥)(𝑥)=1=𝑣⋅𝑢ℎ(𝑥)𝐶₁𝑥+𝐶₂. (2.11)

3. Applications and Examples

Example 3.1. Solve the equation: 𝑑𝛼𝑑𝑥1+𝛽𝑥^𝛾𝑦^′+𝛼𝛽²𝛾𝛾−241+𝛽𝑥^𝛾−2𝑦=0, (3.1) where𝛼,𝛽,𝛾are constants and 𝛼≠0 [7–9].

Solution 1. By virtue of (1.1), we have ℎ(𝑥)=𝛼1+𝛽𝑥^𝛾,𝜓(𝑥)=𝛼𝛽²𝛾𝛾−241+𝛽𝑥^𝛾−2. (3.2) Obviously, that (2.9) is established, so we have: 𝑦(𝑥)(𝑥)(𝑥)=1=𝑣⋅𝑢𝛼1+𝛽𝑥^𝛾𝐶₁𝑥+𝐶₂. (3.3)

Example 3.2. Solve the equation: 𝑑𝑑𝑥𝛼𝑒^𝛾𝑥𝑦^′+𝛼𝛾²4𝑒^𝛾𝑥𝑦=0, (3.4) where 𝛼,𝛾 are constants and 𝛼≠0 [7–9].

Solution 2. By virtue of (1.1), we have ℎ(𝑥)=𝛼⋅𝑒^𝛾𝑥,𝜓(𝑥)=𝛼𝛾²4𝑒^𝛾𝑥. (3.5) Obviously, that (2.9) is established, so we have: 𝑦(𝑥)(𝑥)(𝑥)=1=𝑣⋅𝑢√𝛼𝑒^𝛾𝑥𝐶₁𝑥+𝐶₂. (3.6)

Example 3.3. Solve the equation: 𝑑𝑑𝑥𝛼⋅𝑥^𝑛𝑦^′+𝛼⋅𝑛⋅(𝑛−2)4𝑥^𝑛−2𝑦=0, (3.7) where 𝛼,𝑛 are constants and 𝛼≠0 [7–9].

Solution 3. By virtue of (1.1), (2.9), and (2.11), we have 𝑦(𝑥)(𝑥)(𝑥)=1=𝑣⋅𝑢√𝛼⋅𝑥^𝑛𝐶₁𝑥+𝐶₂. (3.8)

4. Conclusion

The governing equation for stability analysis of a variable cross-section bar subject to variably distributed axial loads, dynamic analysis of multi-storey building, tall building, and other systems is written in the form of a unified self-adjoint equation of the second order. These are reduced to Bessel’s equation in this paper.

The key step in transforming the unified equation to self-adjoint equation is the selection of ℎ(𝑥) and 𝜓(𝑥) in (1.1).

Many difficult problems in the field of static and dynamic mechanics are solved by the unified equation proposed in this paper.