A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem

and Applied Analysis 3 where Y x [ y x z x ] , A x [ 0 1 r x − λ 0 ] , C0 [ 1 0 0 0 ] , C1 [ 0 0 1 0 ] . 2.2 From Ashyralyev and Sobolevskii 8 , we will consider the application of Taylor’s decomposition of function Y x on two points. We need to find Y j x for any 1 ≤ j ≤ p and q. Using the equation Y ′ x A x Y x , we get Y j x Aj x Y x , 2.3 with A0 x I, A1 x A x , Aj x Aj−1 x Aj−1 x A x , 2 ≤ j ≤ p, 2.4 where I is the 2 × 2 identity matrix. By using the structure of the matrix A x , we obtain the entries of the matrix of Aj x [ aj 1,1 λ;x aj 1,2 λ;x aj 2,1 λ;x aj 2,2 λ;x ] 2.5 as in the following formulas: aj 1,1 λ;x ∂aj−1 1,1 λ;x ∂x r x − λ aj−2 2,2 λ;x aj−1 2,1 λ;x , aj 2,2 λ;x ∂aj−1 2,2 λ;x ∂x aj 1,1 λ;x , aj 1,2 λ;x aj−1 2,2 λ;x , aj 2,1 λ;x − ∂aj 2,2 λ;x ∂x aj 1 2,2 λ;x , 2.6 for 2 ≤ j ≤ p, where a0 1,1 λ;x 1, a1 1,1 λ;x 0, a0 1,2 λ;x 0, a1 1,2 λ;x 1, a0 2,1 λ;x 0, a1 2,1 λ;x r x − λ, a0 2,2 λ;x 1, a1 2,2 λ;x 0. 2.7 4 Abstract and Applied Analysis From the theorem given in Ashyralyev and Sobolevskii 8 , we have the following relation: Y xk − Y xk−1 p ∑ j 1 αjY j xk h − q ∑ j 1 βjY j xk−1 h −1 p ( p q ) ! ∫xk xk−1 xk − s q s − xk−1 Y p q 1 s ds 2.8 on the uniform grid x0, xn h {xk x0 kh, k 0, 1, . . . , n, nh xn − x0, n ∈ N}, 2.9


Introduction
We investigate the computation of eigenvalues of regular Sturm-Liouville eigenvalue problems: −y x r x y x λy x , 0 ≤ x 0 < x < x n , y x 0 y x n 0, 1.1 where r x ∈ C p q x 0 , x n and p, q ∈ N and Euler Buckling problem: y λ sin y 0, y 0 0, y 1 0.

1.2
Regular Sturm-Liouville problems arise in many applications, and many methods are available for their numerical solution Pryce 1 .
We also examine an elementary, classical problem buckling of an end-loaded rod which possesses a completely soluble continuous model in the form of a nonlinear, secondorder boundary value problem as described in elsewhere 2-5 .An essential complete analysis of this problem was provided by Euler 6 .For the nonlinear eigenvalue problem 1.2 , one may find that for small λ the only solution is zero solution as in the linear case.But as the eigenvalue λ increases, it reaches a critical value λ 1 at which a nonzero solution appears, corresponding to buckling of the rod.For λ > λ 1 , the nonlinear problem behaves quite differently from the linear problem: for a range of values λ 1 < λ < λ 2 , there is exactly one nonzero solution of 1.2 for each λ, and when λ exceeds λ 2 , a second nonzero solution appears; similarly, there is a value λ 3 beyond which there are three nonzero solutions, and so on.Namely, one may give inductively 0 ≤ λ ≤ π 2 , only the trivial solution, π 2 < λ ≤ 4π 2 , one nontrivial solution, n 2 π 2 < λ ≤ n 1 2 π 2 , n nontrivial solutions, 1.3 as given by Stakgold 2 .This behavior is a simple example of the phenomenon of bifurcation or branching; it occurs in many different areas of applied mathematics.
The method considered here is a Taylor decomposition which was used by Adiyaman and Somali 7 for the solution of certain nonlinear problems.Like classical finite-difference and finite-element methods, this high order method is best suited to the fundamental eigenvalue and small eigenvalues.
In Section 2, the behavior of eigenvalues and corresponding eigenfunctions for regular Sturm-Liouville problem is obtained by Taylor's decomposition method, and convergence of the method for regular Sturm-Liouville problem with constant function r x is given.We establish a Lemma and a Theorem, and then we give an application of Taylor's decomposition method to the Euler Buckling problem in Section 3. The technique is illustrated with three examples, and the numerical results of regular Sturm-Liouville problem are given by comparing the results of other methods in Section 4. The numerical results of Euler Buckling problem accompanying the theoretical results and the behavior of solution are also discussed in Section 4. In the conclusion, we summarize the study and present our suggestions regarding future work.

Application of Taylor's Decomposition on Two Points for Regular Sturm-Liouville Eigenvalue Problems
We consider the regular Sturm-Liouville eigenvalue problem 1.1 by introducing the new depending variable y x z x , 1.1 can be written as

2.1
Abstract and Applied Analysis 3 where From Ashyralyev and Sobolevskii 8 , we will consider the application of Taylor's decomposition of function Y x on two points.We need to find Y j x for any 1 ≤ j ≤ p and q.Using the equation Y x A x Y x , we get where I is the 2 × 2 identity matrix.By using the structure of the matrix A x , we obtain the entries of the matrix of A j x a j 1,1 λ; x a j 1,2 λ; x a j 2,1 λ; x a j 2,2 λ; x 2.5 as in the following formulas: for 2 ≤ j ≤ p, where

2.7
From the theorem given in Ashyralyev and Sobolevskii 8 , we have the following relation: on the uniform grid where

2.10
Rewriting 2.8 by neglecting the last term, we obtain the single-step difference scheme of p q -order of accuracy for the approximate solution of problem 2.1 : where is the approximate value of Y x k .For the simple computation, let p q, then we have where β j −1 j α j .Letting M x k I p j 1 α j A j x k h j and N x k−1 I p j 1 −1 j α j A j x k−1 h j , we write

2.14
Since the accuracy and convergence of the method not only depend on h, they also depend on p, we can increase the order of accuracy by increasing p for fixed h.So h is chosen as length of the whole interval as follows.Now, taking h x n − x 0 gives and substituting into the boundary condition of 2.1 , we get To obtain a nontrivial solution Y 0 , we must have the following equation: det Since

2.20
using the entries m 12 , m 22 , n 12 , and n 22 of the above matrices and the properties of the entries of A j x , we obtain 2.19 in terms of λ:

2.21
Solving nonlinear equation F λ 0 by Newton's method, we find the approximate eigenvalues.This method appears to require a separate calculation for the eigenfunctions.
To find the corresponding eigenfunctions of the regular Sturm-Liouville eigenvalue problem 2.1 , we substitute the eigenvalue to 2.1 and we solve the obtained boundary value problem by Taylor's decomposition method on two points x k−1 and x k with the uniform grid 0, 1 h for p q.Then, we get a homogeneous linear equation system of 2n equations with 2n unknown z 0 , y 1 , z 1 , y 2 , z 2 . . ., y n−1 , z n−1 , z n which are the approximated values of y x 0 , y x 1 , y x 1 , y x 2 , y x 2 , . . ., y x n−1 , y x n−1 , y x n , respectively.Solving the 2n×2n homogeneous system, we obtain approximate values of the eigenfunction and its derivative of 1.1 at the point x x k .

Error Analysis for Regular Sturm-Liouville Problem When r x c
In this section, we will show the convergence of the method for eigenfunctions with the constant function r x c by obtaining approximate value of eigenfunction at the point x ∈ x 0 , x n of the problem 1.1 .Without loss of generality, we may choose r x 0, then A j x A j , that is, a j 2,2 λ; x n a j 2,2 λ; 0 a j 2,2 λ .Using 2.6 , we can find explicit values of a j 1,1 , a j 2,2 as follows:

2.22
This yields

2.23
Using 2.14 for k 1, we have where Y 0 and Y 1 are the approximated values of Y x 0 and Y x , respectively, with the stepsize h x − x 0 :

2.25
The first component of the above vector 2.25 gives the approximate eigenfunction y 1 , and the second component of the above vector 2.25 gives the derivative of the approximate eigenfunction z 1 of the regular Sturm-Liouville problem 1.1 at x. Now, we will show that y 1 and z 1 converge to exact functions y x and y x , respectively, as p → ∞.
Using the Stirling's Formula n! ≈ √ 2πn n 1 /2 e −n for α j in 2.10 , we obtain

2.26
This gives

2.28
By using the same idea, we obtain

2.29
It follows from 2.28 and 2.29 that

2.30
Hence, for r x 0, the approximate eigenfunction of 1.1 to the corresponding eigenvalue λ converges to exact eigenfunction: Since we have z x y x , the derivative of approximate eigenfunction of 1.1 to the corresponding eigenvalue λ converges to derivative of the exact solution: where λ k 2 π 2 , k 1, 2, . ... This demonstration shows that approximate eigenfunction and eigenvalue converges to exact one as p → ∞ for fixed step-size "h." Abstract and Applied Analysis 9

Taylor's Decomposition Method to the Euler Buckling Problem
For convenience, we introduce the following notations as in 2.1 and Adiyaman and Somali 7 : −λ sin y.

2.33
Thus, the Euler Buckling Problem 1.2 can be written in the form:

2.36
We first give the following lemma which defines f j−1 2 y, z explicitly. where

10
Abstract and Applied Analysis for m 0, . . ., p, and where

2.41
for y 1 −y 0 , b it holds that p j 1

2.45
Substituting the value z 0 into 2.45 , we obtain which gives the following relations: Using 2.47 for y 1 −y 0 , we obtain the following relations:

2.48
Similarly for y 1 y 0 using 2.47 , we observe that

2.49
So, our assertions a and b are proved.
Again, we consider the application of Taylor's decomposition method to 2.34 on two points x k and x k−1 : where Y j k is the approximate value of Y j k x k .For the computation of the eigenvalues of 1.2 , putting h 1 and p q, the approximation 2.50 gives   where α j −1 j β j .Writing 2.51 with respect to the components and imposing the boundary conditions z 0 z 0 y 0 0 and z 1 z 1 y 1 0, we have the following equations Using Theorem 2.2 a for y 1 −y 0 , 2.52 becomes and 2.53 is satisfied.For y 1 y 0 , 2.52 is satisfied by Theorem 2.2 b and 2.53 becomes

2.55
From Table 1, we observe that there is only trivial initial condition for 0 ≤ λ ≤ π 2 , there is one nontrivial initial condition from 2.54 for π 2 < λ ≤ 4π 2 , there are n nontrivial initial conditions for n 2 π 2 < λ ≤ n 1 2 π 2 .These results show that the numerical results obtained using Taylor's decomposition method agree with the theoretical results of Euler buckling problem given in 2 .Now, we find an approximate solution to the problem

2.56
Which corresponds to Euler buckling problem 1.2 for an eigenvalue λ and the initial value y 0 .Using Taylor's Decomposition on two points x k−1 , x k for p q then y 0 y 0 ,   z 0 z 0 .Solving the obtained nonlinear system by Newton's method, we obtain the approximate value y k of the eigenfunction y x at Using the results Adiyaman and Somali 7, Lemma 2 and Theorem 3 , the global error for 2.50 is bounded by where , and B h L p j 1 β j h j−1 for some x > 0.

Numerical Results for Regular Sturm-Liouville Eigenvalue Problems
We consider three regular Sturm-Liouville eigenvalue problems, one of them has polynomial coefficients and the others have periodic coefficients taken from Bujurke et al. 9 and Andrew 10 .
Example 3.1.Consider the Titchmarch equation: where n is a nonnegative integer.We obtain the numerical solutions taking n 0, 2. The accuracy of the method is tested by comparing with the exact solution which exists when n 0 and finite-difference method FDM solution and Haar wavelet series method HWSM solution when n 2.
Tables 2 and 3 give computed eigenvalues and solution y x of Titchmarch problem using Taylor's decomposition method TDM with different values of p, HWSM and FDM for n 0, 2, the integer parameter in Titchmarch problem.In Table 2, it is easily seen that the error between approximate eigenfunction and exact eigenfunction decreases as p increases or the step-size decreases or both happen.So, we can find good approximation to eigenfunctions for relatively large step-sizes by increasing p.In Table 3 m is the number of intervals.Table 4 gives the errors between exact and approximate eigenvalues for fixed step-size h 1 for n 0. Notice that, as p increases, the accuracy of approximation almost doubles in digits which demonstrates a fast convergence.

3.2
We will solve these two problems approximately using Taylor's decomposition method TDM , and we will compare our results with the results in Bujurke et al. 9 .Bujurke et al. 9 solved Examples 3.1 and 3.2 approximately using Haar wavelets.They transform

3.3
The eigenvalues for a fixed value for θ 5 are obtained in Table 6 which gives the asymptotic behavior of higher eigenvalues of Mathieu's equation, and these eigenvalues are λ n n 2 O 1 .This result agrees with the classical theorem on asymptoticity of the eigenvalues lim n → ∞ λ 1/2 n /n 1 from van Brunt 11 .Figure 1 demonstrates that the nth eigenfunction has n − 1 zeros in 0,1 which is consistent with the relevant graph in Bujurke et al. 9 .The selected values of parameter θ shifts the symmetry of the solutions and this property is given in Figure 2.

3.4
We give the comparison of approximate eigenvalues obtained using Taylor's Decomposition method with the approximate eigenvalues obtained using Numerov's method Andrew 10 for β 10 and β 20 in Tables 7 and 8.The values shown as the "exact" λ k and the corrected approximate eigenvalues Λ k obtained using Numerov's method for step-sizes h 40 and h 80 in Tables 7 and 8 are taken from Andrew 10 .The values shown as μ k are evaluated using Taylor's Decomposition method for p 70 and 80 in Table 7 and for p 100 and 110 in Table 8.From the tables, it can be seen that Taylor's Decomposition method approximates small eigenvalues with high-order accuracy without using any correction.
In comparison to Example 3.1, the estimation of eigenvalues for Examples 3.2 and 3.3 is more complicated.But Example 3.1 is important to show the high accuracy of the method while calculating the eigenfunctions for relatively large step-sizes.Other two examples show the accuracy of the method while calculating the eigenvalues for large step-sizes which equal to whole interval.In Table 5, the observed orders ord h are computed using the following formula ord h log y 4h − y 2h / y 2h − y h log 2 , 3.5 where y 4h , y 2h , and y h are the approximated value of eigenfunctions at x k to the corresponding eigenvalue λ when the problems are solved with step sizes 4h, 2h, and h respectively.The observed orders given in Table 5 well confirm the theoretical results.That is, the order of TDM is order of 2p.
The numerical calculations and all figures in this work are performed using Mathematica.

Conclusion
In this paper, we have described Taylor's Decomposition method for regular Sturm-Liouville eigenvalue problems with Dirichlet and Neumann boundary conditions to obtain approximate eigenvalues and eigenfunctions and for Euler Buckling Problem to obtain approximate initial values and eigenfunctions.The obtained results for Euler Buckling problem give the behavior of eigenvalues and corresponding eigenfunctions with highorder accuracy without using small stepsize.We have seen that these results agree with the theoretical aspects.This method can be extended to solve regular Sturm-Liouville eigenvalue problems with Robin mixed boundary conditions and to some nonlinear eigenvalue problems to investigate the behavior of the eigenvalues and eigenfunction.However, this method is best suited to find small eigenvalues for the other nonlinear problems in literature.
One possible method of improving its efficiency for higher eigenvalues may be to follow the ideas of 10, 12, 13 and for eigenvalue problems for partial differential equations given in elsewhere 14-18 .

Table 2 :
Comparison of the first eigenvalue and solutions of Example 3.1 using Taylor's decomposition method, exact values, and Table4from 9 , when n 0.

Table 3 :
Comparison of the first eigenvalue and solutions of Example 3.1 using TDM and Table 4 from 9 , when p 16, n 2, and h 0.0625.

Table 4 :
The errors between exact and approximate fundamental eigenvalue for various p and for h 1 and n 0 in Example 3.1.

Table 5 :
Observed orders of Example 3.1 for n 2 at x 1/2 using Taylor's decomposition method.

Table 6 :
Comparison of higher eigenvalues for Mathieu's equation obtained from FDM, HWSM, and TDM corresponding to θ 5.

Table 7 :
Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov's method Λ k with correction and TDM μ k for Example 3.3 corresponding to β 10.

Table 8 :
Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov's method Λ k with correction and TDM μ k for Example 3.3 corresponding to β 20.