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August  2020, 19(8): 4143-4158. doi: 10.3934/cpaa.2020185

Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula

1. 

Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Saudi Arabia

2. 

Institute of Mathematics, University of Lübeck, D-23562 Lübeck, Germany

* Corresponding author

Received  September 2019 Revised  February 2020 Published  May 2020

Fund Project: The first author gratefully acknowledges the support by the Alexander von Humboldt Foundation under grant 3.4-YEM/1142916

The use of sampling methods in computing eigenpairs of two-parameter boundary value problems is extremely rare. As far as we know, there are only two studies up to now using the bivariate version of the classical and regularized sampling series. These series have a slow convergence rate. In this paper, we use the bivariate sinc-Gauss sampling formula that was proposed in [6] to construct a new sampling method to compute eigenpairs of a two-parameter Sturm-Liouville system. The convergence rate of this method will be of exponential order, i.e. $ O(\mathrm{e}^{-\delta N}/\sqrt{N}) $ where $ \delta $ is a positive number and $ N $ is the number of terms in the bivariate sinc-Gaussian formula. We estimate the amplitude error associated to this formula, which gives us the possibility to establish the rigorous error analysis of this method. Numerical illustrative examples are presented to demonstrate our method in comparison with the results of the bivariate classical sampling method.

Citation: Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
References:
[1]

A. AlAzemiF. AlAzemi and A. Boumenir, The approximation of eigencurves by sampling, Sampl. Theory Signal Image Process., 12 (2013), 127-138. 

[2]

M. H. Annaby and R. M. Asharabi, Computing eigenvalues of boundary-value problems using sinc-Gaussian method, Sampl. Theory Signal Image Process., 7 (2008), 293-311. 

[3]

M. H. Annaby and R. M. Asharabi, Computing eigenvalues of Sturm-Liouville problems by Hermite interpolations, Numer. Algor., 60 (2012), 355-367.  doi: 10.1007/s11075-011-9518-x.

[4]

R. M. Asharabi, A Hermite-Gauss technique for approximating eigenvalues of regular Sturm-Liouville problems, J. Inequal. Appl., (2016), Art. 154. doi: 10.1186/s13660-016-1098-9.

[5]

R. M. Asharabi, Generalized bivariate Hermite-Gauss sampling, Comput. Appl. Math., 38 (2019), 29. doi: 10.1007/s40314-019-0802-z.

[6]

R. M. Asharabi and J. Prestin, On two-dimensional classical and Hermite sampling, IMA J. Numer. Anal., 36 (2016), 851-871.  doi: 10.1093/imanum/drv022.

[7]

R. M. Asharabi and M. Tharwat, The use of the Generalized sin-Gaussian sampling for numerically computing eigenvalues of Dirac system, Electron. Trans. Numer. Anal., 48 (2018), 373-386.  doi: 10.1553/etna_vol48s373.

[8]

F. V. Atkinson and A. B. Mingarelli, Multiparameter Eigenvalue Problems, Sturm-Liouville Theory, Vol. 2, CRC Press, Boca Raton, 2011.

[9]

P. A. Binding and P. J. Browne, Asymptotics of eigencurves for second order ordinary differential equations Ⅰ, J. Differ. Equ., 88 (1990), 30-45.  doi: 10.1016/0022-0396(90)90107-Z.

[10]

P. A. Binding and P. J. Browne, Asymptotics of eigencurves for second order ordinary differential equations Ⅱ, J. Differ. Equ., 89 (1991), 224-243.  doi: 10.1016/0022-0396(91)90120-X.

[11]

P. A. Binding and H. Volkmer, Eigencurves for two-parameter Sturm-Liouville equations, SIAM Rev., 38 (1996), 27-48.  doi: 10.1137/1038002.

[12]

P. A. Binding and B. A. Watson, An inverse nodal problem for two-parameter Sturm-Liouville systems, Inverse Probl., 25 (2009), Art. 085005. doi: 10.1088/0266-5611/25/8/085005.

[13]

A. Boumenir and B. Chanane, Eigenvalues of Sturm-Liouville systems using sampling theory, Appl. Anal., 62 (1996), 323-334.  doi: 10.1080/00036819608840486.

[14]

B. Chanane, Computation of eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using reularized sampling method, Math. Comput., 74 (2005), 1793-1801.  doi: 10.1090/S0025-5718-05-01717-5.

[15]

B. Chanane and A. Boucherif, Computation of the eigenpairs of two-parameter Sturm-Liouville problems using the regularized sampling method, Abstr. Appl. Anal., (2014), Art. 695303. doi: 10.1155/2014/695303.

[16]

M. Faierman, The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ordinary differential equations, J. Differ. Equ., 5 (1969), 197-213.  doi: 10.1016/0022-0396(69)90112-0.

[17]

M. R. Sampford, Some inequalities on Mill's ratio and related functions, Ann. Math. Stat., 24 (1953), 130-132.  doi: 10.1214/aoms/1177729093.

show all references

References:
[1]

A. AlAzemiF. AlAzemi and A. Boumenir, The approximation of eigencurves by sampling, Sampl. Theory Signal Image Process., 12 (2013), 127-138. 

[2]

M. H. Annaby and R. M. Asharabi, Computing eigenvalues of boundary-value problems using sinc-Gaussian method, Sampl. Theory Signal Image Process., 7 (2008), 293-311. 

[3]

M. H. Annaby and R. M. Asharabi, Computing eigenvalues of Sturm-Liouville problems by Hermite interpolations, Numer. Algor., 60 (2012), 355-367.  doi: 10.1007/s11075-011-9518-x.

[4]

R. M. Asharabi, A Hermite-Gauss technique for approximating eigenvalues of regular Sturm-Liouville problems, J. Inequal. Appl., (2016), Art. 154. doi: 10.1186/s13660-016-1098-9.

[5]

R. M. Asharabi, Generalized bivariate Hermite-Gauss sampling, Comput. Appl. Math., 38 (2019), 29. doi: 10.1007/s40314-019-0802-z.

[6]

R. M. Asharabi and J. Prestin, On two-dimensional classical and Hermite sampling, IMA J. Numer. Anal., 36 (2016), 851-871.  doi: 10.1093/imanum/drv022.

[7]

R. M. Asharabi and M. Tharwat, The use of the Generalized sin-Gaussian sampling for numerically computing eigenvalues of Dirac system, Electron. Trans. Numer. Anal., 48 (2018), 373-386.  doi: 10.1553/etna_vol48s373.

[8]

F. V. Atkinson and A. B. Mingarelli, Multiparameter Eigenvalue Problems, Sturm-Liouville Theory, Vol. 2, CRC Press, Boca Raton, 2011.

[9]

P. A. Binding and P. J. Browne, Asymptotics of eigencurves for second order ordinary differential equations Ⅰ, J. Differ. Equ., 88 (1990), 30-45.  doi: 10.1016/0022-0396(90)90107-Z.

[10]

P. A. Binding and P. J. Browne, Asymptotics of eigencurves for second order ordinary differential equations Ⅱ, J. Differ. Equ., 89 (1991), 224-243.  doi: 10.1016/0022-0396(91)90120-X.

[11]

P. A. Binding and H. Volkmer, Eigencurves for two-parameter Sturm-Liouville equations, SIAM Rev., 38 (1996), 27-48.  doi: 10.1137/1038002.

[12]

P. A. Binding and B. A. Watson, An inverse nodal problem for two-parameter Sturm-Liouville systems, Inverse Probl., 25 (2009), Art. 085005. doi: 10.1088/0266-5611/25/8/085005.

[13]

A. Boumenir and B. Chanane, Eigenvalues of Sturm-Liouville systems using sampling theory, Appl. Anal., 62 (1996), 323-334.  doi: 10.1080/00036819608840486.

[14]

B. Chanane, Computation of eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using reularized sampling method, Math. Comput., 74 (2005), 1793-1801.  doi: 10.1090/S0025-5718-05-01717-5.

[15]

B. Chanane and A. Boucherif, Computation of the eigenpairs of two-parameter Sturm-Liouville problems using the regularized sampling method, Abstr. Appl. Anal., (2014), Art. 695303. doi: 10.1155/2014/695303.

[16]

M. Faierman, The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ordinary differential equations, J. Differ. Equ., 5 (1969), 197-213.  doi: 10.1016/0022-0396(69)90112-0.

[17]

M. R. Sampford, Some inequalities on Mill's ratio and related functions, Ann. Math. Stat., 24 (1953), 130-132.  doi: 10.1214/aoms/1177729093.

Figure 1.  The eigencurves in Example 1
Figure 2.  (a) The logarithm of the norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,20}, \mu_{k,0,20})\|_{\mathbb{R}^{2}} $ for $ k = 1,\ldots,6 $ in Example 1. (b) The logarithm of the norm error $ \|(\lambda_{3}^{*},\mu_{3}^{*})-(\lambda_{3,0,N},\mu_{3,0,N})\|_{\mathbb{R}^{2}} $ for $ N = 10,15,20,25 $ in Example 1
Figure 3.  The eigencurves in Example 2
Figure 4.  The eigencurves in Example 3
Table 1.  Comparisons
Methods Region of approximation Convergence rate
WKS sampling $ [-N,N]^{2} $ $ \ln N/\sqrt{N} $
Regularized sampling $ [-N,N]^{2} $ $ \ln N/N^{m+1/2} $
Sinc-Gaussian sampling $ \prod_{j=1}^{2}[(n_{j}-1/2)h_{j},(n_{j}+1/2)h_{j}] $ $ \mathrm{e}^{-\delta N}/\sqrt{N} $
Methods Region of approximation Convergence rate
WKS sampling $ [-N,N]^{2} $ $ \ln N/\sqrt{N} $
Regularized sampling $ [-N,N]^{2} $ $ \ln N/N^{m+1/2} $
Sinc-Gaussian sampling $ \prod_{j=1}^{2}[(n_{j}-1/2)h_{j},(n_{j}+1/2)h_{j}] $ $ \mathrm{e}^{-\delta N}/\sqrt{N} $
Table 2.  Approximation of eigenpairs with $ h = 1 $
$ k $ $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Bivariate WKS sampling
1 1.813797507802172 1.513239555736101
2 3.627597850186581 3.487076569018237
3 5.441403076170987 5.348799863878748
4 7.255208077727408 7.186073252549332
5 9.069006371615362 9.013757735420938
6 10.88280054723645 10.836602869705539
Bivariate sinc-Gauss sampling
1 1.813799364683959 1.513231023664942
2 3.627598728958227 3.487043523167927
3 5.441398093112097 5.348720707354780
4 7.255197457187725 7.185950886348564
5 7.255197457187725 9.013695321137181
6 10.882796185506988 10.83675471755794
$ k $ $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Bivariate WKS sampling
1 1.813797507802172 1.513239555736101
2 3.627597850186581 3.487076569018237
3 5.441403076170987 5.348799863878748
4 7.255208077727408 7.186073252549332
5 9.069006371615362 9.013757735420938
6 10.88280054723645 10.836602869705539
Bivariate sinc-Gauss sampling
1 1.813799364683959 1.513231023664942
2 3.627598728958227 3.487043523167927
3 5.441398093112097 5.348720707354780
4 7.255197457187725 7.185950886348564
5 7.255197457187725 9.013695321137181
6 10.882796185506988 10.83675471755794
Table 3.  The norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,15},\mu_{k,0,15})\|_{\mathbb{R}^{2}} $
$ k $ Bivariate WKS sampling Bivariate sinc-Gauss sampling
$ h=1 $ $ h=0.5 $
1 8.73082$ \times 10^{-6} $ 1.00333$ \times 10^{-9} $ 6.94810$ \times 10^{-12} $
2 3.30572$ \times 10^{-5} $ 5.91952$ \times 10^{-10} $ 1.23606$ \times 10^{-11} $
3 7.93134$ \times 10^{-5} $ 4.50801$ \times 10^{-10} $ 3.57188$ \times 10^{-11} $
4 1.22826$ \times 10^{-4} $ 2.82605$ \times 10^{-10} $ 2.22197$ \times 10^{-11} $
5 6.31407$ \times 10^{-5} $ 2.54946$ \times 10^{-10} $ 2.27273$ \times 10^{-12} $
6 1.51911$ \times 10^{-4} $ 2.33596$ \times 10^{-10} $ 7.64573$ \times 10^{-12} $
$ k $ Bivariate WKS sampling Bivariate sinc-Gauss sampling
$ h=1 $ $ h=0.5 $
1 8.73082$ \times 10^{-6} $ 1.00333$ \times 10^{-9} $ 6.94810$ \times 10^{-12} $
2 3.30572$ \times 10^{-5} $ 5.91952$ \times 10^{-10} $ 1.23606$ \times 10^{-11} $
3 7.93134$ \times 10^{-5} $ 4.50801$ \times 10^{-10} $ 3.57188$ \times 10^{-11} $
4 1.22826$ \times 10^{-4} $ 2.82605$ \times 10^{-10} $ 2.22197$ \times 10^{-11} $
5 6.31407$ \times 10^{-5} $ 2.54946$ \times 10^{-10} $ 2.27273$ \times 10^{-12} $
6 1.51911$ \times 10^{-4} $ 2.33596$ \times 10^{-10} $ 7.64573$ \times 10^{-12} $
Table 4.  Approximation of eigenpairs with $ h = 1 $
{$ k $} $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Bivariate WKS sampling
1 1.359821568195881 1.584365124779384
2 2.294869574533618 3.950449753437753
3 5.235270180088456 2.129614898841256
4 6.477462750322421 4.683407147326487
5 7.667060430768335 6.932768257873039
6 8.825101408478323 9.106626719982193
Bivariate sinc-Gauss sampling
1 1.359811348447286 1.584379611568847
2 2.294859272608290 3.950445447061610
3 5.235258286227501 2.129611255383138
4 6.477390507161662 4.683473685527605
5 7.666946848415846 6.932950168963926
6 8.825213824551480 9.106473269299752
{$ k $} $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Bivariate WKS sampling
1 1.359821568195881 1.584365124779384
2 2.294869574533618 3.950449753437753
3 5.235270180088456 2.129614898841256
4 6.477462750322421 4.683407147326487
5 7.667060430768335 6.932768257873039
6 8.825101408478323 9.106626719982193
Bivariate sinc-Gauss sampling
1 1.359811348447286 1.584379611568847
2 2.294859272608290 3.950445447061610
3 5.235258286227501 2.129611255383138
4 6.477390507161662 4.683473685527605
5 7.666946848415846 6.932950168963926
6 8.825213824551480 9.106473269299752
Table 5.  The norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,15},\mu_{k,0,15})\|_{\mathbb{R}^{2}} $ with $ h = 1 $
$ k $ Bivariate WKS sampling Bivariate sinc-Gauss sampling
1 1.77230$ \times 10^{-5} $ 6.16607$ \times 10^{-9} $
2 1.11696$ \times 10^{-5} $ 8.90886$ \times 10^{-9} $
3 1.24318$ \times 10^{-5} $ 7.90168$ \times 10^{-9} $
4 9.82270$ \times 10^{-5} $ 1.60926$ \times 10^{-8} $
5 2.14480$ \times 10^{-4} $ 2.13712$ \times 10^{-8} $
6 1.90214$ \times 10^{-4} $ 8.33743$ \times 10^{-9} $
$ k $ Bivariate WKS sampling Bivariate sinc-Gauss sampling
1 1.77230$ \times 10^{-5} $ 6.16607$ \times 10^{-9} $
2 1.11696$ \times 10^{-5} $ 8.90886$ \times 10^{-9} $
3 1.24318$ \times 10^{-5} $ 7.90168$ \times 10^{-9} $
4 9.82270$ \times 10^{-5} $ 1.60926$ \times 10^{-8} $
5 2.14480$ \times 10^{-4} $ 2.13712$ \times 10^{-8} $
6 1.90214$ \times 10^{-4} $ 8.33743$ \times 10^{-9} $
Table 6.  Approximation of eigenpairs with $ h = 1 $ and $ \varepsilon = 10^{-8} $
{$ k $} $ \lambda_{k,\varepsilon,15} $ $ \mu_{k,\varepsilon,15} $
[1ex] Bivariate WKS sampling
1 0.515656277786066 0.762177530812667
2 2.051784932534724 2.114071060975086
3 3.478736174942723 3.516120586250193
4 4.893280857549082 4.919999452785837
5 6.303637655607843 6.324376642474745
6 7.712016456380302 7.728978003793143
7 9.119198704295004 9.133705129467295
8 10.525522861700562 10.538515646451458
Bivariate sinc-Gauss sampling
1 0.515671212590693 0.762173604088073
2 2.051799490194234 2.114068514591409
3 3.478721556147922 3.516123605356794
4 4.893200190982574 4.920011111879944
5 6.303486384409280 6.324392419791860
6 7.711851697945461 7.728986314987891
7 9.119176716920121 9.133693454689087
8 10.525875454418745 10.538468065160703
{$ k $} $ \lambda_{k,\varepsilon,15} $ $ \mu_{k,\varepsilon,15} $
[1ex] Bivariate WKS sampling
1 0.515656277786066 0.762177530812667
2 2.051784932534724 2.114071060975086
3 3.478736174942723 3.516120586250193
4 4.893280857549082 4.919999452785837
5 6.303637655607843 6.324376642474745
6 7.712016456380302 7.728978003793143
7 9.119198704295004 9.133705129467295
8 10.525522861700562 10.538515646451458
Bivariate sinc-Gauss sampling
1 0.515671212590693 0.762173604088073
2 2.051799490194234 2.114068514591409
3 3.478721556147922 3.516123605356794
4 4.893200190982574 4.920011111879944
5 6.303486384409280 6.324392419791860
6 7.711851697945461 7.728986314987891
7 9.119176716920121 9.133693454689087
8 10.525875454418745 10.538468065160703
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