American Institute of Mathematical Sciences

Advanced
  • 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

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

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.

Keywords: Sinc approximation, multiparameter spectral theory, eigencurve, Sturm-Liouville systems, Bernstein space.
Mathematics Subject Classification: Primary: 34B05, 34B09; Secondary: 65F18.
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:
[1]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The eigencurves in Example 1
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The eigencurves in Example 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The eigencurves in Example 3
Figure Options

Download full-size image

Download as PowerPoint slide

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} $
Table Options

Download as excel

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
Table Options

Download as excel

$ 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} $
Table Options

Download as excel

$ 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
Table Options

Download as excel

{$ 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} $
Table Options

Download as excel

$ 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
Table Options

Download as excel

{$ 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]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
[2]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[3]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
[4]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[5]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
[6]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
[7]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145
[8]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
[9]

Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171
[10]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[11]

Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[12]

Diogo A. Gomes, Gabriele Terrone. Bernstein estimates: weakly coupled systems and integral equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 861-883. doi: 10.3934/cpaa.2012.11.861
[13]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083
[14]

Lei Wang, Meijun Zhu. Liouville theorems on the upper half space. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5373-5381. doi: 10.3934/dcds.2020231
[15]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041
[16]

Harun Karsli. On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators. Mathematical Foundations of Computing, 2021, 4 (1) : 15-30. doi: 10.3934/mfc.2020023
[17]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249
[18]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[19]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[20]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (91)
  • HTML views (78)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences