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

Rashad M. Asharabi; Jürgen Prestin

doi:10.3934/cpaa.2020185

Article Contents

2020, Volume 19, Issue 8: 4143-4158. Doi: 10.3934/cpaa.2020185

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

Rashad M. Asharabi^1, and
Jürgen Prestin^2, ,

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
^* Corresponding author

Received: August 31, 2019

Revised: January 31, 2020

Published: May 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34B05, 34B09; Secondary: 65F18.

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Figure 1. The eigencurves in Example 1

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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

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Figure 3. The eigencurves in Example 2

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Figure 4. The eigencurves in Example 3

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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} $

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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

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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
$ k $	Bivariate WKS 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} $

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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

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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} $

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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

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