# American Institute of Mathematical Sciences

• Previous Article
Kernel-based maximum correntropy criterion with gradient descent method
• CPAA Home
• This Issue
• Next Article
First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $\alpha$-stable white noise
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 & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
##### References:

show all references

##### References:
The eigencurves in Example 1
(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
The eigencurves in Example 2
The eigencurves in Example 3
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 $\prod_{j=1}^{2}[(n_{j}-1/2)h_{j},(n_{j}+1/2)h_{j}]$ $\mathrm{e}^{-\delta N}/\sqrt{N}$
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
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$\times 10^{-12}$
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
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$\times 10^{-9}$
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
 [1] Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083 [2] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 [3] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [4] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 [5] John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004 [6] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [7] Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016 [8] Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005 [9] Enkhbat Rentsen, N. Tungalag, J. Enkhbayar, O. Battogtokh, L. Enkhtuvshin. Application of survival theory in Mining industry. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 443-448. doi: 10.3934/naco.2020036 [10] Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389 [11] Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006 [12] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [13] Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021025 [14] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395 [15] Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021008 [16] W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 [17] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020 [18] Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076 [19] Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 [20] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

2019 Impact Factor: 1.105