Asymptotic and Numerical Methods in Estimating Eigenvalues

Abstract

Asymptotic formulas and numerical estimations for eigenvalues of SturmLiouville problems having singular potential functions, with Dirichlet boundary conditions, are obtained. This study gives a comparison between the eigenvalues obtained by the asymptotic and the numerical methods.

1. Introduction

Let 𝐿(𝑞) be an operator generated in 𝐿₂[0,1] by the expression 𝐿𝑞=−𝑦^′′(𝑥)+𝑞(𝑥)𝑦(𝑥),0≤𝑥≤1, (1) and by Dirichlet boundary conditions 𝑦(0)=𝑦(1)=0, (2) where 𝑞(𝑥) is a complex-valued summable function.

In this paper, we consider the small and large eigenvalues of the operator 𝐿(𝑞) when 𝑞(𝑥) has a finite number of singularities. The large eigenvalues are investigated by the asymptotic method given in [1, 2]. Note that in classical investigations in order to obtain the asymptotic formulas of order 𝑂(𝑛^−𝑙) it is required that 𝑞(𝑥) be (𝑙−1) times differentiable (see [3–10]). The method of [1] gives the possibility of obtaining the asymptotic formulas of order 𝑂(𝑛^−𝑙) of eigenvalues and eigenfunctions of 𝐿(𝑞) when 𝑞(𝑥) is an arbitrary summable complex-valued function. The small eigenvalues are investigated by numerical and asymptotic methods. Then, we compare the results with the ones obtained by the other methods.

Expression of differential equations in matrix form and the advances in the field of the computers have led to major developments in numerical methods. Regarding the numerical solution of the Sturm-Liouville problems, finite difference method is amongst the popular methods (see [11, 12]). Finite difference method can give effective results for the eigenvalues when it is used in connection with asymptotic correction technique. In [13] and [14] the Sturm-Liouville problems with Dirichlet and the general boundary conditions were studied, respectively. Andrew and Paine [15] found the approximate eigenvalues of regular Sturm-Liouville problem by using the finite element method. Chen and Ho [16] used the differential transform method to solve the eigenvalue problems. Ghelardoni [17] named some linear multistep methods as boundary value methods and found the approximate eigenvalues of Sturm-Liouville problem. Ghelardoni and Gheri [18] used the shooting technique for the calculation of the eigenvalues of Sturm-Liouville problem by considering the Prüfer transformation given in [19]. Kumar [20], Kumar and Aziz [21] gave numerical examples to linear or nonlinear boundary value problems by using finite differences method for singular boundary value problems. Kumar and Singh [22] made a study which collected and classified various calculation techniques for the solution of singular boundary value problems.

2. Asymptotic Formulas for Eigenvalues

It is well known that (see formulas (47a), (47b) in page 65 of [7]) the eigenvalues of the operator 𝐿(𝑞), where 𝑞(𝑥) is a complex-valued summable function, consist of the sequence {𝜆_𝑛} satisfying 𝜆_𝑛=(𝑛𝜋)²+𝑂(1). (3)

In [1] (see Theorem 1 of [1]), it is proved that the eigenvalues 𝜆_𝑛 satisfy the following formula 𝜆_𝑛=(𝑛𝜋)²+𝐶₀+𝐹_𝑚|𝑛|+𝑂ln𝑛^𝑚+1,∀𝑚=0,1,2,…, (4) where 𝐹₀=−𝐶_2𝑛, 𝐶_𝑛=∫¹₀𝑞(𝑥)cos𝑛𝜋𝑥𝑑𝑥, 𝐹₁=𝐴₁(𝑛𝜋)²=−𝐶_2𝑛+^∞_{𝑛1𝑛=−∞1≠−2𝑛}𝐶_𝑛1𝐶_𝑛1−𝐶_𝑛1+2𝑛(𝑛𝜋)²−𝜋𝑛+𝑛₁²,𝐹_𝑘=𝐴_𝑘(𝑛𝜋)²+𝐹_𝑘−1,∀𝑘=2,3,…. (5)(6) Note that in [1], without loss of generality, it was assumed that 𝐶₀=0. Then using (4), the cases 𝑞(𝑥)=𝑝(𝑥)+𝑐/𝑥^𝛼 and 𝑞(𝑥)=𝑎₀/√𝑥+𝑎₁/√1−𝑥+𝑏₀√𝑥+𝑏₁√1−𝑥 (where 𝑐, 𝑎₀, 𝑎₁, 𝑏₀, 𝑏₁ are complex numbers) are investigated in detail.

In this paper, we consider the case 𝑞(𝑥)=^𝜈_𝑘=0𝑐_𝑘𝑥−𝑡_𝑘^𝛼𝑘,0<𝛼_𝑘<1, (7) where 𝜈 is a positive integer and 𝑐_𝑘 is a complex number. First using (4) we prove the following.

Theorem 1. The eigenvalue 𝜆_𝑛 of the operator 𝐿(𝑞) with potential (7) satisfies the asymptotic formula: 𝜆_𝑛=(𝑛𝜋)²+𝐶₀−^𝜈_𝑘=0𝑐_𝑘(2𝑛)^1−𝛼𝑘cos2𝑛𝜋𝑡_𝑘𝑑_4𝑘+𝑑_4𝑘+2−sin2𝑛𝜋𝑡_𝑘𝑑_4𝑘+1+𝑑_4𝑘+3|𝑛|+𝑂ln𝑛, (8) where 𝑑_4𝑘=^∞₀cos𝜋𝑠𝑠^𝛼𝑘𝑑𝑠,𝑑_4𝑘+2=⁰_−∞cos𝜋𝑠𝑠^𝛼𝑘𝑑𝑑𝑠,_4𝑘+1=⁰_−∞sin𝜋𝑠𝑠^𝛼𝑘𝑑𝑠,𝑑_4𝑘+3=^∞₀sin𝜋𝑠𝑠^𝛼𝑘𝑑𝑠. (9)

Proof. At (4) for 𝑚=0, let us use the formula 𝜆_𝑛=(𝜋𝑛)²+𝐶₀−𝐶_2𝑛|𝑛|+𝑂ln𝑛, (10) where 𝐶_𝑛=¹₀𝑞(𝑥)cos𝑛𝜋𝑥𝑑𝑥=^𝜈_𝑘=0¹₀𝑐_𝑘cos𝑛𝜋𝑥𝑥−𝑡_𝑘^𝛼𝑘𝑑𝑥. (11) In the last equality, using the transformations 𝑡=𝑥−𝑡_𝑘 and 𝑛𝑡=𝑠 we obtain 𝐶_𝑛=^𝜈_𝑘=0^1−𝑡𝑘_−𝑡𝑘𝑐_𝑘cos𝑛𝜋𝑡+𝑡_𝑘𝑡^𝛼𝑘_𝑘=𝑑𝑡_{𝜈𝑘=01−𝑡𝑘−𝑡𝑘}𝑐_𝑘cos𝑛𝜋𝑡cos𝑛𝜋𝑡_𝑘𝑡^𝛼𝑘_𝑘−𝑑𝑡_{𝜈𝑘=01−𝑡𝑘−𝑡𝑘}𝑐_𝑘sin𝑛𝜋𝑡sin𝑛𝜋𝑡_𝑘𝑡^𝛼𝑘_𝑘𝐶𝑑𝑡,_𝑛=^𝜈_𝑘=0𝑐_𝑘𝑛^1−𝛼𝑘cos𝑛𝜋𝑡_𝑘^{𝑛(1−𝑡𝑘)}_{−𝑛𝑡𝑘}cos𝜋𝑠𝑠^𝛼𝑘𝑑𝑠−sin𝑛𝜋𝑡_𝑘^{𝑛(1−𝑡𝑘)}_{−𝑛𝑡𝑘}sin𝜋𝑠𝑠^𝛼𝑘𝑑𝑠. (12) Let 𝑛(1−𝑡_𝑘)=𝑛_𝑘. By (9) we have 𝐶_𝑛=^𝜈_𝑘=0𝑐_𝑘𝑛^1−𝛼𝑘cos𝑛𝜋𝑡_𝑘𝑑_4𝑘−^∞_𝑛𝑘cos𝜋𝑠𝑠^𝛼𝑘𝑑𝑠+𝑑_4𝑘+2−^{−𝑛𝑡𝑘}_−∞cos𝜋𝑠𝑠^𝛼𝑘𝑑𝑠−sin𝑛𝜋𝑡_𝑘𝑑_4𝑘+1−^{−𝑛𝑡𝑘}_−∞sin𝜋𝑠𝑠^𝛼𝑘𝑑𝑠+𝑑_4𝑘+3−^∞_𝑛𝑘sin𝜋𝑠𝑠^𝛼𝑘𝑑𝑠. (13) Arguing as in the proof of (41) of [1] one can readily see that ^{−𝑛𝑡𝑘}_−∞cos𝜋𝑠𝑠^𝛼𝑘1𝑑𝑠=𝑂𝑛^𝛼𝑘_𝑘,^∞_𝑛𝑘cos𝜋𝑠𝑠^𝛼𝑘1𝑑𝑠=𝑂𝑛^𝛼𝑘_{𝑘−𝑛𝑡𝑘−∞},sin𝜋𝑠𝑠^𝛼𝑘1𝑑𝑠=𝑂𝑛^𝛼𝑘_𝑘,^∞_𝑛𝑘sin𝜋𝑠𝑠^𝛼𝑘1𝑑𝑠=𝑂𝑛^𝛼𝑘_𝑘. (14) Therefore 𝐶_𝑛=^𝜈_𝑘=0𝑐_𝑘𝑛^1−𝛼𝑘cos𝑛𝜋𝑡_𝑘𝑑_4𝑘+𝑑_4𝑘+2−sin𝑛𝜋𝑡_𝑘𝑑_4𝑘+1+𝑑_4𝑘+31+𝑂𝑛. (15) Thus (8) follows from (4) for 𝑚=0. The theorem is proved.

Now assuming that 𝛼₁=𝛼₂=⋯=𝛼_𝜈=12 (16) we obtain more precise asymptotic formula by using more subtle estimations.

Theorem 2. If (16) holds, then the eigenvalue 𝜆_𝑛 of the operator 𝐿(𝑞) with potential (7) satisfies the asymptotic formula: 𝜆_𝑛=(𝑛𝜋)²+𝐶₀−𝐶_2𝑛|𝑛|+𝑂ln𝑛². (17)

Proof. To prove the theorem we use (4) for 𝑚=1, (5) and prove that ^∞_{𝑛1𝑛=−∞1≠0,−2𝑛}𝐶²_𝑛1𝑛₁2𝑛+𝑛₁|𝑛|=𝑂ln𝑛²_{∞𝑛1𝑛=−∞1≠0,−2𝑛},𝐶_𝑛1𝐶_𝑛1+2𝑛𝑛₁2𝑛+𝑛₁|𝑛|=𝑂ln𝑛². (18)
In (15), instead of 𝑛 and 𝛼_𝑘 taking 𝑛₁ and 1/2 we get 𝐶_𝑛1=^𝜈_𝑘=0𝑐_𝑘𝑛₁^1/2cos𝑛₁𝜋𝑡_𝑘𝑑_4𝑘+𝑑_4𝑘+2−sin𝑛₁𝜋𝑡_𝑘𝑑_4𝑘+1+𝑑_4𝑘+31+𝑂𝑛₁. (19) From (19) one can readily see that there exists a constant 𝑟 such that 𝐶_𝑛1²1<𝑟𝑛₁ (20) for 𝑛₁=1,2,…. Therefore, instead of equation (56) of [1], using (20) and repeating the proof of equation (55) of [1] we get the proof of (18). Thus the proof of the theorem follows from (4), (5), and (18). The theorem is proved.

3. Numerical Approximation

Now, we consider the small eigenvalues of the 𝐿(𝑞) operator by a numerical method.

For the finite difference method [11, 19] take an equally spaced mesh (𝑚⩾2)0=𝑥₀<𝑥₁<⋯<𝑥_𝑚+1=1, (21) where 𝑥_𝑗1=𝑗ℎ,ℎ=.𝑚+1 (22) Writing 𝑦(𝑥_𝑗) as 𝑦_𝑗, 𝑞(𝑥_𝑗) as 𝑞_𝑗, and 𝑦^′′(𝑥_𝑗) as 𝑦_𝑗^′′, we use the centered difference approximation −𝑦_𝑗^′′≈−𝑦_𝑗−1+2𝑦_𝑗−𝑦_𝑗+1ℎ². (23) Substituting in (1) we obtain the approximating scheme −𝑌_𝑗−1+2𝑌_𝑗−𝑌_𝑗+1ℎ²+𝑞_𝑗𝑌_𝑗=Λ𝑌_𝑗,𝑗=1,2,…,𝑚. (24) Incorporating the boundary conditions, we get 𝑌₀=0,𝑌_𝑚+1=0. (25) This can be written in matrix form as 𝑇𝑌=Λ𝑌, (26) where 1𝑇=ℎ²𝐾+𝑄^∗, (27)𝑇 is a tridiagonal matrix and 𝑌𝑌=¹⋅⋅⋅𝑌_𝑚,⋅⋅⋅,𝑄𝐾=2−1−12−1−12−1−12_∗=𝑞₁𝑞₂⋅⋅⋅𝑞_𝑚. (28) The eigenvalues of (1), (2) are approximated by the eigenvalues of matrix 𝑇.

In the previous section, the asymptotic formulas for eigenvalues of the operator 𝐿(𝑞) (1), (2) with the potential (7) are investigated. In this section, we will find the eigenvalues of the operator 𝐿(𝑞) by using the finite difference method when 𝑐_𝑘=1, 𝛼_𝑘=1/2 for 𝑘=0,1,…,𝜈, and 𝑡_𝑘=𝑘/𝜈. Let us introduce the notation 𝑞_𝑘(𝑥)=1𝑥−𝑡_𝑘^1/2 (29) and denote the 𝑛th eigenvalue of the operator 𝐿(𝑞_𝑘) by 𝜆^𝑘_𝑛. The 𝑛th eigenvalue of the operator 𝐿(𝑄_𝜈), where 𝑄_𝜈(𝑥)=^𝜈_𝑘=0𝑞_𝑘(𝑥), (30) is denoted by Λ^𝜈_𝑛.

In order to be able to apply the Finite Difference method, the nodes should not coincide with the singular points. Let 𝑚_𝜈=𝜈(𝑚+1) and 𝑥_𝑗 nodal points be 𝑥_𝑗=𝑗𝑚_𝜈,𝑗=1,2,…,𝑚_𝜈−1;𝑗≠𝑘(𝑚+1). (31) Then 𝑥_𝑗≠𝑡_𝑘.

The approximate eigenvalues of the operators 𝐿(𝑞_𝑘) and 𝐿(𝑄_𝜈) obtained by the numerical method are denoted 𝑠^𝑘_𝑛 and 𝑆^𝜈_𝑛, respectively.

Example 3. In this example we find the eigenvalues of the following boundary value problem −𝑦^′′+⁵_𝑘=01𝑥−𝑡_𝑘^1/2𝑦(0)(1)𝑦=𝜆𝑦,=𝑦=0 (32) for𝑚_𝜈=150, 𝑡_𝑘=𝑘/𝜈, and 𝜈=5 by using Finite Difference method. In Table 1 an example of the computation of the eigenvalues of the operators 𝐿(𝑞_𝑘) and 𝐿(𝑄_𝜈) is given.
One can see from Table 1 that for 𝑛≥30 the eigenvalues of the operators 𝐿(𝑞_𝑘) and 𝐿(𝑄_𝑣) are close to each other. This shows that the effect of the potential to the large eigenvalues is small. Moreover the eigenvalues in first, second, and third columns coincide with the eigenvalues in the sixth, fifth, and fourth columns, respectively, since the potential 𝑞_𝑘(𝑥) can be reduced to 𝑞_𝜈−𝑘(𝑥) by using the transformation 𝑥=1−𝑧.

Table 1 

4. Comparison of the Asymptotic and Numerical Methods

In this section we compare the estimations obtained by numerical and asymptotic methods of the eigenvalues Λ²_𝑛 of the operator 𝐿(𝑄₂), where 𝑄₂ is defined by (30) and (29). The 𝑛th eigenvalue of the operator 𝐿(0) is (𝑛𝜋)². The effect of the potential 𝑞_𝑘 on the 𝑛th eigenvalue 𝜆^𝑘_𝑛 of the operator 𝐿(𝑞_𝑘), that is, the perturbation of the 𝑛th eigenvalue when 𝐿(0) is perturbed by 𝑞_𝑘 is 𝑝^𝑘_𝑛=𝜆^𝑘_𝑛−(𝑛𝜋)². (33) Similarly, the effect of 𝑄₂ on the 𝑛th eigenvalue Λ²_𝑛, that is, the perturbation of the 𝑛th eigenvalue when 𝐿(0) is perturbed by 𝑄₂ is 𝑃²_𝑛=Λ²_𝑛−(𝑛𝜋)². (34) The perturbations 𝑃²_𝑛, 𝑝^𝑘_𝑛 evaluated by the numerical and asymptotic methods are denoted by 𝑃²_𝑛(𝑠), 𝑝^𝑘_𝑛(𝑠), 𝑃²_𝑛(𝑎), and 𝑝^𝑘_𝑛(𝑎), respectively.

According to Theorem 2 we define the approximate eigenvalues, denoted by 𝑎^𝑘_𝑛 and 𝐴²_𝑛, of the operators 𝐿(𝑞_𝑘) and 𝐿(𝑄₂) obtained by the asymptotic method as follows 𝑎^𝑘_𝑛=(𝑛𝜋)²+¹₀𝑞_𝑘(𝑥)𝑑𝑥−¹₀𝑞_𝑘(𝑥)𝐴cos2𝜋𝑛𝑥𝑑𝑥,_2𝑛=(𝑛𝜋)²+¹₀𝑄₂(𝑥)𝑑𝑥−¹₀𝑄₂(𝑥)cos2𝜋𝑛𝑥𝑑𝑥. (35)(36) Therefore it is natural to define 𝑃²_𝑛(𝑎) and 𝑝^𝑘_𝑛(𝑎) by 𝑃²_𝑛(𝑎)=𝐴²_𝑛−(𝑛𝜋)²=¹₀𝑄₂(𝑥)𝑑𝑥−¹₀𝑄₂(𝑥)𝑝cos2𝜋𝑛𝑥𝑑𝑥,_𝑘𝑛(𝑎)=𝑎^𝑘_𝑛−(𝑛𝜋)²=¹₀𝑞_𝑘(𝑥)𝑑𝑥−¹₀𝑞_𝑘(𝑥)cos2𝜋𝑛𝑥𝑑𝑥. (37) It readily follows from formulas (37) and (30) that 𝑃²_𝑛(𝑎)=²_𝑘=0𝑝^𝑘_𝑛(𝑎). (38) It means that for the large eigenvalues the effect of 𝑄₂ is asymptotically equal to the sum of the effects of the potentials 𝑞_𝑘.

The perturbations 𝑃²_𝑛(𝑠) and 𝑝^𝑘_𝑛(𝑠) evaluated via the finite difference method are given in Table 2. In order to see the effect of the singular points, the number of subintervals is taken as 𝑚_𝜈=20000.

Table 2 

Table 2 shows that the effect of 𝑄₂ is approximately within the value range of 5.10⁻⁵ and 2.10⁻², equal to the sum of the effects of the potentials 𝑞_𝑘. Thus the perturbation estimations by the numerical methods validate the naturality of (38).

It is well known that if we consider the Sturm-Liouville operator 𝐿𝜀𝑄₂𝑑=−𝑑𝑥²+𝜀𝑄₂(𝑥),𝑦(0)=𝑦(1)=0, (39) where 𝜀 is a small positive parameter, then the asymptotic methods can be applied more successfully. The 𝑛th eigenvalue of the operators 𝐿(𝜀𝑄_𝜈) is denoted by Λ^𝜈_𝑛(𝜀). The approximate eigenvalues obtained by the asymptotic and numerical methods are denoted by 𝐴^𝜈_𝑛(𝜀) and 𝑆^𝜈_𝑛(𝜀), respectively.

It follows from Theorem 2 and formulas (36), (30) that Λ²_𝑛(𝜀)=𝐴²_𝑛(𝜀)|𝑛|+𝑂𝜀ln𝑛², (40) where 𝐴²_𝑛(𝜀)=(𝜋𝑛)²+𝜀¹₀𝑄₂(𝑥)𝑑𝑥−𝜀𝐶_2𝑛=(𝜋𝑛)²+𝜀¹₀1||+1𝑥−1||+1𝑥−1/2|𝑥|𝑑𝑥−𝜀¹₀1||+1𝑥−1||+1𝑥−1/2|𝑥|cos2𝑛𝜋𝑥𝑑𝑥. (41)

In Tables 3, 4, and 5 the approximate eigenvalues 𝐴²_𝑛(𝜀) obtained by the asymptotic method and their comparison with 𝑆²_𝑛(𝜀) and nonperturbated eigenvalues 𝜆_𝑛(0)=(𝑛𝜋)² for 𝜀=1, 𝜀=0,1, and 𝜀=0,01, respectively, are given.

Table 3 

Table 4 

Table 5 

Table 3 shows the eigenvalues of 𝐿(𝜀𝑄₂) operator obtained by asymptotic method and finite difference method, respectively, for 𝜀=1. Here the number of subintervals is taken as 𝑚_𝜈=15000.

Table 4 shows the eigenvalues of 𝐿(𝜀𝑄₂) operator obtained by asymptotic method and finite difference method, respectively, for 𝜀=0,1. Here the number of subintervals is taken as 𝑚_𝜈=15000.

Table 5 shows the eigenvalues of 𝐿(𝜀𝑄₂) operator obtained by asymptotic method and finite difference method, respectively, for 𝜀=0,01. Here the number of subintervals is taken as 𝑚_𝜈=15000.

5. Conclusion

It is natural and well known that for small values of the parameter 𝜀 and for large eigenvalues the asymptotic method gives us approximations with smaller errors. The numerical method, in general, gives better results for smaller eigenvalues. The tables show that the results of the asymptotic method also give quiet acceptable results for small eigenvalues, since 𝐴²_𝑛(𝜀)−𝑆²_𝑛(𝜀) is small.

Therefore we can easily observe that both of two methods give high-precision results for the calculation of the small eigenvalues. Additionally while the perturbation parameter tends to zero both of the methods are enhanced for smaller eigenvalues, but while this fact is limited to 𝑛=10 for the numerical approximation, the enhancement continues for the asymptotic method applied to higher eigenvalues. Thus we can conclude that the asymptotic method coupled with a perturbation parameter near to zero provides us a better approximation quality in calculating eigenvalues.

In Tables 3–5 there are two observations to be considered: the first observation is that for small eigenvalues the perturbated results by numerical and asymptotic methods are close to each other for all 𝜀. The second observation is that for the large eigenvalues the perturbated results obained by asymptotic methods decrease linearly with respect to small 𝜀, while the perturbated results obtained by numerical methods are almost the same for all values of 𝜀. This shows that for small values of the perturbation parameter 𝜀 the asymptotic method is preferable.

Conflict of Interests

The authors of the paper do not have any direct or indirect financial relation with the commercial identities mentioned in the paper.