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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 |
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 [
References:
[1] |
A. AlAzemi, F. 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. AlAzemi, F. 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. |




Methods | Region of approximation | Convergence rate |
WKS sampling | ||
Regularized sampling | ||
Sinc-Gaussian sampling |
Methods | Region of approximation | Convergence rate |
WKS sampling | ||
Regularized sampling | ||
Sinc-Gaussian sampling |
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 |
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 |
Bivariate WKS sampling | Bivariate sinc-Gauss sampling | ||
1 | 8.73082 |
1.00333 |
6.94810 |
2 | 3.30572 |
5.91952 |
1.23606 |
3 | 7.93134 |
4.50801 |
3.57188 |
4 | 1.22826 |
2.82605 |
2.22197 |
5 | 6.31407 |
2.54946 |
2.27273 |
6 | 1.51911 |
2.33596 |
7.64573 |
Bivariate WKS sampling | Bivariate sinc-Gauss sampling | ||
1 | 8.73082 |
1.00333 |
6.94810 |
2 | 3.30572 |
5.91952 |
1.23606 |
3 | 7.93134 |
4.50801 |
3.57188 |
4 | 1.22826 |
2.82605 |
2.22197 |
5 | 6.31407 |
2.54946 |
2.27273 |
6 | 1.51911 |
2.33596 |
7.64573 |
{ |
||
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 |
{ |
||
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 |
Bivariate WKS sampling | Bivariate sinc-Gauss sampling | |
1 | 1.77230 |
6.16607 |
2 | 1.11696 |
8.90886 |
3 | 1.24318 |
7.90168 |
4 | 9.82270 |
1.60926 |
5 | 2.14480 |
2.13712 |
6 | 1.90214 |
8.33743 |
Bivariate WKS sampling | Bivariate sinc-Gauss sampling | |
1 | 1.77230 |
6.16607 |
2 | 1.11696 |
8.90886 |
3 | 1.24318 |
7.90168 |
4 | 9.82270 |
1.60926 |
5 | 2.14480 |
2.13712 |
6 | 1.90214 |
8.33743 |
{ |
||
[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 |
{ |
||
[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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