# Asymptotic and Numerical Methods in Estimating Eigenvalues

Authors: Guldem Yıldız; Bulent Yılmaz; O. A. Veliev
Journal: Mathematical Problems in Engineering
 (2013)

---

## 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.


---

## Body

## 1. Introduction

LetL(q) be an operator generated in L2[0,1] by the expression
(1)L(q)=-y′′(x)+q(x)y(x),0≤x≤1,
and by Dirichlet boundary conditions
(2)y(0)=y(1)=0,
where q(x) is a complex-valued summable function.In this paper, we consider the small and large eigenvalues of the operatorL(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 {λn} satisfying
(3)λn=(nπ)2+O(1).In [1] (see Theorem 1 of [1]), it is proved that the eigenvalues λn satisfy the following formula
(4)λn=(nπ)2+C0+Fm+O((ln|n|n)m+1),0+O+O((ln|n|n)m+1)∀m=0,1,2,…,
where F0=-C2n, Cn=∫01‍q(x)cosnπxdx,
(5)F1=A1((nπ)2)=-C2n+∑n1=-∞n1≠-2n∞Cn1(Cn1-Cn1+2n)[(nπ)2-(π(n+n1))2],(6)Fk=Ak((nπ)2+Fk-1),∀k=2,3,….
Note that in [1], without loss of generality, it was assumed that C0=0. Then using (4), the cases q(x)=p(x)+c/xα and q(x)=a0/x+a1/1-x+b0x+b11-x (where c, a0, a1, b0, b1 are complex numbers) are investigated in detail.In this paper, we consider the case(7)q(x)=∑k=0ν‍ck|x-tk|αk,0<αk<1,
where ν is a positive integer and ck is a complex number. First using (4) we prove the following.Theorem 1.
The eigenvalueλn of the operator L(q) with potential (7) satisfies the asymptotic formula:
(8)λn=(nπ)2+C0-∑k=0ν‍ck(2n)1-αk(cos2nπtk(d4k+d4k+2)-sin2nπtk(d4k+1+d4k+3))+O(ln|n|n),
where
(9)d4k=∫0∞‍cosπssαkds,d4k+2=∫-∞0‍cosπssαkds,d4k+1=∫-∞0sinπssαkds,d4k+3=∫0∞‍sinπssαkds.Proof.
At (4) for m=0, let us use the formula
(10)λn=(πn)2+C0-C2n+O(ln|n|n),
where
(11)Cn=∫01‍q(x)cosnπxdx=∑k=0ν‍∫01‍ckcosnπx(x-tk)αkdx.
In the last equality, using the transformations t=x-tk and nt=s we obtain
(12)Cn=∑k=0ν‍∫-tk1-tk‍ckcosnπ(t+tk)tkαkdt=∑k=0ν‍∫-tk1-tk‍ckcosnπtcosnπtktkαkdt-∑k=0ν‍∫-tk1-tk‍cksinnπtsinnπtktkαkdt,Cn=∑k=0ν‍ckn1-αk(cosnπtk∫-ntkn(1-tk)‍cosπssαkds-sinnπtk∫-ntkn(1-tk)‍sinπssαkds).
Let n(1-tk)=nk. By (9) we have
(13)Cn=∑k=0ν‍ckn1-αk(cosnπtk(d4k-∫nk∞‍cosπssαkds+d4k+2-∫-∞-ntk‍cosπssαkds)-sinnπtk(d4k+1-∫-∞-ntk‍sinπssαkds+d4k+3-∫nk∞‍sinπssαkds)).
Arguing as in the proof of  (41) of [1] one can readily see that
(14)∫-∞-ntk‍cosπssαkds=O(1nkαk),∫nk∞‍cosπssαkds=O(1nkαk),∫-∞-ntk‍sinπssαkds=O(1nkαk),∫nk∞‍sinπssαkds=O(1nkαk).
Therefore
(15)Cn=∑k=0ν‍ckn1-αk(cosnπtk(d4k+d4k+2)-sinnπtk(d4k+1+d4k+3))+O(1n).
Thus (8) follows from (4) for m=0. The theorem is proved.Now assuming that(16)α1=α2=⋯=αν=12
we obtain more precise asymptotic formula by using more subtle estimations.Theorem 2.
If (16) holds, then the eigenvalue λn of the operator L(q) with potential (7) satisfies the asymptotic formula:
(17)λn=(nπ)2+C0-C2n+O((ln|n|n)2).Proof.
To prove the theorem we use (4) for m=1, (5) and prove that
(18)∑n1=-∞n1≠0,-2n∞‍Cn12n1(2n+n1)=O((ln|n|n)2),∑n1=-∞n1≠0,-2n∞‍Cn1Cn1+2nn1(2n+n1)=O((ln|n|n)2).
In (15), instead of n and αk taking n1 and 1/2 we get
(19)Cn1=∑k=0ν‍ck(n1)1/2(cosn1πtk(d4k+d4k+2)-sinn1πtk(d4k+1+d4k+3))+O(1n1).
From (19) one can readily see that there exists a constant r such that
(20)|Cn1|2<r1n1
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 theL(q) operator by a numerical method.For the finite difference method [11, 19] take an equally spaced mesh (m⩾2)(21)0=x0<x1<⋯<xm+1=1,
where
(22)xj=jh,h=1m+1.
Writing y(xj) as yj, q(xj) as qj, and y′′(xj) as yj′′, we use the centered difference approximation
(23)-yj′′≈-yj-1+2yj-yj+1h2.
Substituting in (1) we obtain the approximating scheme
(24)-Yj-1+2Yj-Yj+1h2+qjYj=ΛYj,j=1,2,…,m.
Incorporating the boundary conditions, we get
(25)Y0=0,Ym+1=0.
This can be written in matrix form as
(26)TY=ΛY,
where
(27)T=1h2K+Q*,T is a tridiagonal matrix and
(28)Y=(Y1···Ym),K=(2-1-12-1···-12-1-12),Q*=(q1q2···qm).
The eigenvalues of (1), (2) are approximated by the eigenvalues of matrix T.In the previous section, the asymptotic formulas for eigenvalues of the operatorL(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 ck=1, αk=1/2 for k=0,1,…,ν, and tk=k/ν. Let us introduce the notation
(29)qk(x)=1|x-tk|1/2
and denote the nth eigenvalue of the operator L(qk) by λnk. The nth eigenvalue of the operator L(Qν), where
(30)Qν(x)=∑k=0ν‍qk(x),
is denoted by Λnν.In order to be able to apply the Finite Difference method, the nodes should not coincide with the singular points. Letmν=ν(m+1) and xj nodal points be
(31)xj=jmν,j=1,2,…,mν-1;j≠k(m+1).
Then xj≠tk.The approximate eigenvalues of the operatorsL(qk) and L(Qν) obtained by the numerical method are denoted snk and Snν, respectively.Example 3.
In this example we find the eigenvalues of the following boundary value problem(32)-y′′+∑k=05‍1|(x-tk)|1/2y=λy,y(0)=y(1)=0
formν=150, tk=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(qk) and L(Qν) is given.
One can see from Table1 that for n≥30 the eigenvalues of  the operators L(qk) and L(Qv) 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 qk(x) can be reduced to qν-k(x) by using the transformation x=1-z.Table 1
n
s
n
0
s
n
1
s
n
2
s
n
3
s
n
4
s
n
5
S
n
5
1
11,3346
12,0617
13,6112
13,6112
12,0617
11,3346
24,8946
2
41,0082
42,3680
42,5655
42,5655
42,3680
41,0082
54,4909
3
90,3543
92,1439
91,4556
91,4556
92,1439
90,3543
103,912
4
159,389
161,282
161,330
161,330
161,282
159,389
174,822
5
248,089
249,774
249,278
249,278
249,774
248,089
261,654
6
356,418
357,740
357,883
357,883
357,740
356,418
369,832
7
484,332
485,347
486,023
486,023
485,347
484,332
497,594
8
631,775
632,690
632,941
632,941
632,690
631,775
643,821
9
789,683
799,727
800,334
800,334
799,727
789,683
811,972
10
984,984
986,288
986,440
986,440
986,288
984,984
998,400
20
3892,79
3864,330
3894,39
3894,39
3864,330
3892,79
3906,97
30
8599,57
8600,85
8601,04
8601,04
8600,85
8599,57
8612,95
40
14902,3
14903,5
14903,7
14903,7
14903,5
14902,3
14915,3
50
22529,2
22530,6
22530,7
22530,7
22530,6
22529,2
22542,7
60
31151,4
31152,8
31152,8
31152,8
31152,8
31151,4
31165,1
70
40396,8
40398
40398,3
40398,3
40398
40396,8
40410,1
80
49866,8
49867,9
49868,2
49868,2
49867,9
49866,8
49879,9
90
59152,9
59154,3
59154,3
59154,3
59154,3
59152,9
59166,4
100
67854,6
67856,1
67856,1
67856,1
67856,1
67854,6
67868,3


## 4. Comparison of the Asymptotic and Numerical Methods

In this section we compare the estimations obtained by numerical and asymptotic methods of the eigenvaluesΛn2 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 qk on the nth eigenvalue λnk of the operator L(qk), that is, the perturbation of the nth eigenvalue when L(0) is perturbed by qk is
(33)pnk=λnk-(nπ)2.
Similarly, the effect of Q2 on the nth eigenvalue Λn2, that is, the perturbation of the nth eigenvalue when L(0) is perturbed by Q2 is
(34)Pn2=Λn2-(nπ)2.
The perturbations Pn2, pnk evaluated by the numerical and asymptotic methods are denoted by Pn2(s), pnk(s), Pn2(a), and pnk(a), respectively.According to Theorem2 we define the approximate eigenvalues, denoted by ank and An2, of the operators L(qk) and L(Q2) obtained by the asymptotic method as follows
(35)ank=(nπ)2+∫01‍qk(x)dx-∫01‍qk(x)cos2πnxdx,(36)An2=(nπ)2+∫01‍Q2(x)dx-∫01‍Q2(x)cos2πnxdx.
Therefore it is natural to define Pn2(a) and pnk(a) by
(37)Pn2(a)=An2-(nπ)2=∫01‍Q2(x)dx-∫01‍Q2(x)cos2πnxdx,pnk(a)=ank-(nπ)2=∫01‍qk(x)dx-∫01‍qk(x)cos2πnxdx.
It readily follows from formulas (37) and (30) that
(38)Pn2(a)=∑k=02‍pnk(a).
It means that for the large eigenvalues the effect of Q2 is asymptotically equal to the sum of the effects of the potentials qk.The perturbationsPn2(s) and pnk(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 mν=20000.Table 2
n
P
n
2
(
s
)
p
n
0
(
s
)
p
n
1
(
s
)
p
n
2
(
s
)
∑
k
=
0
2
p
n
k
(
s
)
|
p
n
2
(
s
)
-
∑
k
=
0
2
p
n
k
(
s
)
|
1
6,853010948
1,507680075
3,819522987
1,507680075
6,834883138
0,018127810
2
5,436436039
1,649044169
2,135885787
1,649044169
5,433974124
0,002461915
3
6,833127542
1,712792234
3,412323071
1,712792234
6,837907540
0,004779998
4
5,833596917
1,750983257
2,332560213
1,750983257
5,834526726
0,000929810
5
6,821354115
1,777106646
3,269640370
1,777106646
6,823853662
0,002499546
6
6,014222685
1,796412736
2,422163800
1,796412736
6,014989272
0,000766587
7
6,817074514
1,811422205
3,195628783
1,811422205
6,818473194
0,001398680
8
6,122590551
1,823514738
2,476005854
1,823514738
6,123035330
0,000444779
9
6,814993899
1,833516325
3,148677685
1,833516325
6,815710336
0,000716437
10
6,196675434
1,841954226
2,512821873
1,841954226
6,196730326
0,000054891
20
6,378247333
1,885045278
2,601825385
1,885045278
6,371915940
0,006331392Table2 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 qk. 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(39)L(εQ2)=-ddx2+εQ2(x),y(0)=y(1)=0,
where ε is a small positive parameter, then the asymptotic methods can be applied more successfully. The nth eigenvalue of the operators L(εQν) is denoted by Λnν(ε). The approximate eigenvalues obtained by the asymptotic and numerical methods are denoted by Anν(ε) and Snν(ε), respectively.It follows from Theorem2 and formulas (36), (30) that
(40)Λn2(ε)=An2(ε)+O((εln|n|n)2),
where
(41)An2(ε)=(πn)2+ε∫01‍Q2(x)dx-εC2n=(πn)2+ε∫01‍(1|x-1|+1|x-1/2|+1|x|)dx-ε∫01‍(1|x-1|+1|x-1/2|+1|x|)cos2nπxdx.In Tables3, 4, and 5 the approximate eigenvalues An2(ε) obtained by the asymptotic method and their comparison with Sn2(ε) and nonperturbated eigenvalues λn(0)=(nπ)2 for ε=1, ε=0,1, and ε=0,01, respectively, are given.Table 3
n
λ
n
(
0
)
A
n
2
(
1
)
S
n
2
(
1
)
|
A
n
2
(
1
)
-
S
n
2
(
1
)
|
|
A
n
2
(
1
)
-
λ
n
(
0
)
|
|
S
n
2
(
1
)
-
λ
n
(
0
)
|
1
9,8696
16,779308
16,7201
0,059208
6,909708
6,8505
2
39,4784
44,915438
44,9149
0,000538
5,437038
5,4365
3
88,8264
95,665322
95,6569
0,008422
6,838922
6,8305
4
157,9137
163,748052
163,7473
0,000752
5,834352
5,8336
5
246,7401
253,572371
253,5588
0,013571
6,832271
6,8187
6
355,3058
361,320361
361,3199
0,000461
6,014561
6,0141
7
483,6106
490,44101
490,4249
0,01611
6,83041
6,8143
8
631,6547
637,77751
637,7770
0,00051
6,12281
6,1223
9
799,4379
806,267576
806,2502
0,017376
6,829676
6,8123
10
986,9604
993,157379
993,1570
0,000379
6,196979
6,1966
20
3947,8418
3954,223216
3954,2175
0,005216
6,381416
6,3757
30
8882,6440
8889,107347
8889,0782
0,029347
6,463347
6,4342
40
15791,3670
15797,8793
15797,7870
0,0893
6,51236
6,42
50
24674,0110
24680,55663
24680,3312
0,22663
6,54563
6,3202
60
35530,5758
35537,1461
35536,6786
0,4661
6,5703
6,1028
70
48361,0616
48367,65097
48366,7849
0,87097
6,58937
5,7233
80
63165,4682
63172,073
63170,5955
1,473
6,6048
5,1273
90
79943,7956
79950,41327
79948,0466
2,36327
6,61767
4,251
100
98696,0440
98702,67245
98699,0652
3,60245
6,62845
3,0212Table 4
n
λ
n
(
0
)
A
n
2
(
0,1
)
S
n
2
(
0,1
)
|
A
n
2
(
0,1
)
-
S
n
2
(
0,1
)
|
|
A
n
2
(
0,1
)
-
λ
n
(
0
)
|
|
S
n
2
(
0,1
)
-
λ
n
(
0
)
|
1
9,8696
10,5605748
10,5582
0,0023748
0,6909748
0,6886
2
39,4784
40,0221196
40,0221
1,96 × 10−5
0,5437196
0,5437
3
88,8264
89,5103278
89,5085
0,0018278
0,6839278
0,6821
4
157,9137
158,4971086
158,4971
8,6 × 10−6
0,5834086
0,5834
5
246,7401
247,4233362
247,4214
0,0019362
0,6832362
0,6813
6
355,3058
355,9072187
355,9072
1,87 × 10−5
0,6014187
0,6014
7
483,6106
484,2936551
484,2916
0,0020551
0,6830551
0,681
8
631,6547
632,2669645
632,2668
0,0001645
0,6122645
0,6121
9
799,4379
800,1209185
800,1187
0,0022185
0,6830185
0,6808
10
986,9604
987,580134
987,5798
0,000334
0,619734
0,6194
20
3947,8418
3948,479906
3948,4741
0,005806
0,638106
0,6323
30
8882,6440
8883,2903
8883,2611
0,0291996
0,6463
0,6171
40
15791,3670
15792,01827
15791,9259
0,0923677
0,65127
0,5589
50
24674,0110
24674,66557
24674,4401
0,225465
0,65457
0,4291
60
35530,5758
35531,23287
35530,7654
0,4674694
0,65707
0,1896
70
48361,0616
48361,72051
48360,8544
0,8661055
0,65891
0,2072
80
63165,4682
63166,12865
63164,6511
1,4775505
0,66045
0,8171
90
79943,7956
79944,45741
79942,0907
2,3667109
0,66181
1,7049
100
98696,0440
98696,70685
98693,0996
3,6072546
0,66285
2,9444Table 5
n
λ
n
(
0
)
A
n
2
(
0,01
)
S
n
2
(
0,01
)
|
A
n
2
(
0,01
)
-
S
n
2
(
0,01
)
|
|
A
n
2
(
0,01
)
-
λ
n
(
0
)
|
|
S
n
2
(
0,01
)
-
λ
n
(
0
)
|
1
9,8696
9,93870144
9,9385
0,00020144
0,06910144
0,0689
2
39,4784
39,53278781
39,5328
1,219 × 10−5
0,05438781
0,0544
3
88,8264
88,89482843
88,8946
0,00022843
0,06842843
0,0682
4
157,9137
157,9720142
157,9720
1,423 × 10−5
0,0583142
0,0583
5
246,7401
246,8084326
246,8082
0,00023264
0,0683326
0,0681
6
355,3058
355,3659045
355,3659
4,46 × 10−6
0,0601045
0,0601
7
483,6106
483,6789196
483,6786
0,0003196
0,0683196
0,068
8
631,6547
631,71591
631,7158
0,00010995
0,06121
0,0611
9
799,4379
799,5062527
799,5058
0,00045269
0,0683527
0,0679
10
986,9604
987,0224095
987,0220
0,00040949
0,0620095
0,0616
20
3947,8418
3947,905575
3947,8998
0,00577499
0,063775
0,058
30
8882,6440
8882,708595
8882,6794
0,02919485
0,064595
0,0354
40
15791,3670
15791,43216
15791,3398
0,09236434
0,06516
0,0272
50
24674,0110
24674,07646
24673,8510
0,22545896
0,06546
0,16
60
35530,5758
35530,64155
35530,1740
0,46754647
0,06575
0,4018
70
48361,0616
48361,12746
48360,2614
0,86605935
0,06586
0,8002
80
63165,4682
63165,53422
63164,0567
1,47751533
0,06602
1,4115
90
79943,7956
79943,86183
79941,4951
2,36672503
0,06623
2,3005
100
98696,0440
98696,1103
98692,5031
3,60719526
0,0663
3,5409Table3 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 mν=15000.Table4 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 mν=15000.Table5 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 mν=15000.


## 5. Conclusion

It is natural and well known that for small values of the parameterε 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 An2(ε)-Sn2(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 ton=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 Tables3–5 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.



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*Source: 415479-2013-04-22.xml*