Abstract and Applied Analysis

Volume 2012 (2012), Article ID 697013, 21 pages

doi:10.1155/2012/697013

Research Article

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

Meltem Evrenosoglu Adiyaman and Sennur Somali

Department of Mathematics, Faculty of Science, Dokuz Eylul University, 35160 Tinaztepe, Buca, Izmir, Turkey

Received 12 March 2012; Accepted 11 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Meltem Evrenosoglu Adiyaman and Sennur Somali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We propose a numerical Taylor’s Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high-order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.

1. Introduction

We investigate the computation of eigenvalues of regular Sturm-Liouville eigenvalue problems:

where and and Euler Buckling problem:

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, second-order 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 the only solution is zero solution as in the linear case. But as the eigenvalue increases, it reaches a critical value at which a nonzero solution appears, corresponding to buckling of the rod. For , the nonlinear problem behaves quite differently from the linear problem: for a range of values , there is exactly one nonzero solution of (1.2) for each , and when exceeds , a second nonzero solution appears; similarly, there is a value beyond which there are three nonzero solutions, and so on. Namely, one may give inductively

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 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.

2. Application and Error Analysis of Taylor’s Decomposition Method for Regular Sturm-Liouville Eigenvalue Problems

2.1. 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 , (1.1) can be written as

where

From Ashyralyev and Sobolevskii [8], we will consider the application of Taylor’s decomposition of function on two points. We need to find for any . Using the equation , we get

with

where is the identity matrix. By using the structure of the matrix , we obtain the entries of the matrix of

as in the following formulas:

for , where

From the theorem given in Ashyralyev and Sobolevskii [8], we have the following relation:

on the uniform grid

where

Rewriting (2.8) by neglecting the last term, we obtain the single-step difference scheme of ()-order of accuracy for the approximate solution of problem (2.1):

where

is the approximate value of . For the simple computation, let , then we have

where . Letting and , we write

Since the accuracy and convergence of the method not only depend on , they also depend on , we can increase the order of accuracy by increasing for fixed . So is chosen as length of the whole interval as follows. Now, taking gives

and substituting into the boundary condition of (2.1), we get

To obtain a nontrivial solution , we must have the following equation:

Defining

we have the following statement

Since

using the entries , and of the above matrices and the properties of the entries of , we obtain (2.19) in terms of :

Solving nonlinear equation 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 and with the uniform grid for . Then, we get a homogeneous linear equation system of equations with unknown which are the approximated values of , respectively. Solving the homogeneous system, we obtain approximate values of the eigenfunction and its derivative of (1.1) at the point .

2.2. Error Analysis for Regular Sturm-Liouville Problem When

In this section, we will show the convergence of the method for eigenfunctions with the constant function by obtaining approximate value of eigenfunction at the point of the problem (1.1). Without loss of generality, we may choose , then , that is, . Using (2.6), we can find explicit values of , as follows:

This yields

Using (2.14) for , we have

where and are the approximated values of and , respectively, with the stepsize :

The first component of the above vector (2.25) gives the approximate eigenfunction , and the second component of the above vector (2.25) gives the derivative of the approximate eigenfunction of the regular Sturm-Liouville problem (1.1) at . Now, we will show that and converge to exact functions and , respectively, as .

Using the Stirling’s Formula for in (2.10), we obtain

This gives

Thus,

By using the same idea, we obtain

It follows from (2.28) and (2.29) that

Hence, for , the approximate eigenfunction of (1.1) to the corresponding eigenvalue converges to exact eigenfunction:

Since we have , the derivative of approximate eigenfunction of (1.1) to the corresponding eigenvalue converges to derivative of the exact solution:

where .

This demonstration shows that approximate eigenfunction and eigenvalue converges to exact one as for fixed step-size “.”

2.3. 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]:

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

Defining the following recurrence relations for :

we obtain

We first give the following lemma which defines explicitly.

Lemma 2.1. For , let satisfy the recurrence relation (2.35) with . Then

where

for , and

where

for .

Proof. The proof follows induction argument based on (2.35).

Theorem 2.2. If and , are sufficiently smooth and satisfy (2.36), (2.37), and (2.39) then the following relations hold: (a) it holds that

for , (b) it holds that

for .

Proof. Let for , then becomes

by Lemma 2.1. Since all terms of previous sum contain , for , hence, we get the following equations:

Letting for and using (2.37), we get

Substituting the value into (2.45), we obtain

which gives the following relations:

Using (2.47) for , we obtain the following relations:

Similarly for using (2.47), we observe that

So, our assertions (a) and (b) are proved.

Again, we consider the application of Taylor’s decomposition method to (2.34) on two points and:

where is the approximate value of . For the computation of the eigenvalues of (1.2), putting and , the approximation (2.50) gives

where . Writing (2.51) with respect to the components and imposing the boundary conditions and , we have the following equations

Using Theorem 2.2(a) for , (2.52) becomes

and (2.53) is satisfied. For , (2.52) is satisfied by Theorem 2.2(b) and (2.53) becomes

From Table 1, we observe that there is only trivial initial condition for , there is one nontrivial initial condition from (2.54) for , there are nontrivial initial conditions for . 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

Which corresponds to Euler buckling problem (1.2) for an eigenvalue and the initial value . Using Taylor’s Decomposition on two points , for then , . Solving the obtained nonlinear system by Newton’s method, we obtain the approximate value of the eigenfunction at with .

It is clear that is Lipschitz in in 2-dimensional box . Using the results (Adiyaman and Somali [7, Lemma  2 and Theorem  3]), the global error for (2.50) is bounded by

where , , , is 2-dimensional box in , , , , const is a constant independent of denotes , with , , , , , and for some .

Table 1: Corresponding to the initial values , , , and for various obtained from (2.54) and (2.55).

1.7471
—
—
—

2.8578
1.0092
—
—

3.0718
2.3413
0.3236
—

3.1272
2.7999
2.0239
0.3771

3. Numerical Results

3.1. 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 is a nonnegative integer. We obtain the numerical solutions taking . The accuracy of the method is tested by comparing with the exact solution which exists when and finite-difference method (FDM) solution and Haar wavelet series method (HWSM) solution when .

Tables 2 and 3 give computed eigenvalues and solution of Titchmarch problem using Taylor’s decomposition method (TDM) with different values of , HWSM and FDM for , the integer parameter in Titchmarch problem. In Table 2, it is easily seen that the error between approximate eigenfunction and exact eigenfunction decreases as increases or the step-size decreases or both happen. So, we can find good approximation to eigenfunctions for relatively large step-sizes by increasing . In Table 3 is the number of intervals. Table 4 gives the errors between exact and approximate eigenvalues for fixed step-size for . Notice that, as increases, the accuracy of approximation almost doubles in digits which demonstrates a fast convergence.

Table 2: Comparison of the first eigenvalue and solutions of Example 3.1 using Taylor’s decomposition method, exact values, and Table 4 from [9], when .

HWSM
FDM
TDM
Exact
Errors of TDM
Errors of TDM
Errors of TDM

()
()
() ()
() ()
() ()
() ()

0
0
0
0
0
0
0
0

0.0625
0.27521
0.278599
0.275999
0.275899

—
—

0.125
0.54181
0.541196
0.541196
0.541196

—

0.1875
0.78549
0.785695
0.785695
0.785695

—
—

0.25
1.00482
1
1
1

0.3125
1.17851
1.17588
1.17588
1.17588

—
—

0.375
1.31285
1.30656
1.30656
1.30656

—

0.4375
1.38376
1.38704
1.38704
1.38704

—
—

0.5
1.41103
1.41421
1.41421
1.41421

0

0.5625
1.38376
1.38704
1.38704
1.38704

—
—

0.625
1.31285
1.30656
1.30656
1.30656

—

0.6875
1.17851
1.17588
1.17588
1.17588

—
—

0.75
1.00482
1
1
1

2.06E-11

0.8125
0.78549
0.785695
0.785695
0.785695

—
—

0.875
0.54181
0.541196
0.541196
0.541196

—

0.9375
0.27521
0.275899
0.275999
0.275899

—
—

1
0
0
0
0
0
0
0

(HWSM), 10.8379 (FDM), 10.8696 (TDM), 10.8696 (Exact).

Table 3: Comparison of the first eigenvalue and solutions of Example 3.1 using TDM and Table 4 from [9], when , , and .

HWSM
FDM
TDM

0
0
0
0

0.0625
0.27521
0.27756
0.277563

0.125
0.54181
0.54434
0.544337

0.1875
0.78949
0.78996
0.789953

0.25
1.00485
1.00488
1.00487

0.3125
1.18153
1.18075
1.18074

0.375
1.31286
1.31082
1.31076

0.4375
1.39372
1.38996
1.38994

0.5
1.42102
1.41527
1.41529

0.5625
1.39371
1.38591
1.38598

0.625
1.31285
1.30323
1.30334

0.6875
1.18154
1.18066
1.17081

0.75
1.0048
0.99361
0.993792

0.8125
0.77949
0.77917
0.779357

0.875
0.53481
0.53577
0.535934

0.9375
0.27726
0.27277
0.272878

1
0
0
0

(HWSM), 9.95067 (FDM), 9.98317 (TDM).

Table 4: The errors between exact and approximate fundamental eigenvalue for various and for and in Example 3.1.

2
3
4
6
7
9
10
11

Errors
2.130

Example 3.2. Consider the Mathieu’s equation:

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 the interval to because of the properties of Haar wavelets. So to compare the results we normalize the interval by using , Mathieu’s equation in Example 3.2 transformed into

The eigenvalues for a fixed value for are obtained in Table 6 which gives the asymptotic behavior of higher eigenvalues of Mathieu’s equation, and these eigenvalues are . This result agrees with the classical theorem on asymptoticity of the eigenvalues from van Brunt [11]. Figure 1 demonstrates that the th eigenfunction has zeros in (0,1) which is consistent with the relevant graph in Bujurke et al. [9]. The selected values of parameter shifts the symmetry of the solutions and this property is given in Figure 2.

Figure 1: Higher eigenfunctions of Mathieu’s equation for a fixed parameter .

Figure 2: Solutions of Mathieu’s equation for different parameters of .

Example 3.3. Consider the equation

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 and in Tables 7 and 8. The values shown as the “exact” and the corrected approximate eigenvalues obtained using Numerov’s method for step-sizes and in Tables 7 and 8 are taken from Andrew [10]. The values shown as are evaluated using Taylor’s Decomposition method for and 80 in Table 7 and for 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 are computed using the following formula

where , , and are the approximated value of eigenfunctions at to the corresponding eigenvalue when the problems are solved with step sizes , , and respectively. The observed orders given in Table 5 well confirm the theoretical results. That is, the order of TDM is order of .

The numerical calculations and all figures in this work are performed using Mathematica.

Table 5: Observed orders of Example 3.1 for at using Taylor’s decomposition method.

4.23247
7.90607
10.6574

3.86153
5.22141
8.13719

3.98674
5.93074
7.63932

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

(FDM)
(HWSM)
(TDM)

1
1
−5.7311
−5.4665
−5.79008

2
4
2.0992
2.6161
2.09946

3
9
9.2365
9.4227
9.23633

4
16
16.648
16.3707
16.6482

5
25
25.511
24.1471
25.5108

6
36
36.359
36.6577
36.3589

(HWSM), −5.73115 (FDM), −5.79008 (TDM).

Table 7: Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov’s method () with correction and TDM () for Example 3.3 corresponding to .

1
0.0000000

2
37.7596285

3
37.8059002

4
37.8525995

5
70.5475097

6
92.6538177

7
96.2058159

8
102.254347

for Taylor’s decomposition method.

Table 8: Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov’s method () with correction and TDM () for Example 3.3 corresponding to .

1
0.0000000

2
77.9161943

3
77.9161957

4
77.9161972

5
151.463224

for Taylor’s decomposition method.

3.2. Numerical Results for Euler Buckling Problem

The approximate solutions of Euler Buckling problem for , , , and generated using Taylor’s Decomposition method for step size are illustrated in Figures 3, 4, 5, and 6, respectively.

Figure 3: Solution of (1.2) corresponding to the initial values for .

Figure 4: Solution of (1.2) corresponding to the initial values for .

Figure 5: Solution of (1.2) corresponding to the initial values for .

Figure 6: Solution of (1.2) corresponding to the initial values for .

4. 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 high-order 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].

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