# 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] Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 [2] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [3] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [4] Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048 [5] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [6] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [7] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [8] Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262 [9] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [10] Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 [11] Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020389 [12] 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 - A, 2020  doi: 10.3934/dcds.2020395 [13] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 [14] Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146 [15] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [16] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [17] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [18] Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230 [19] Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048 [20] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

2019 Impact Factor: 1.105