August  2018, 12(4): 1033-1054. doi: 10.3934/ipi.2018043

On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

1. 

School of Mathematics, Southeast University, Shing-Tung Yau Center of Southeast University, Nanjing 211189, China

2. 

Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan

3. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

4. 

Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

* Corresponding author: Jenn-Nan Wang

Received  October 2017 Revised  February 2018 Published  June 2018

Fund Project: Li is supported in parts by the NSFC 11471074. Huang was partially supported by the Ministry of Science and Technology (MOST) 105-2115-M-003-009-MY3, National Center of Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST, NCTS and ST Yau Center in Taiwan. Wang was partially supported by MOST 105-2115-M-002-014-MY3.

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of $O(1)$ and the other half of eigenvalues are positive with order of $O(10^2)$. In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

Citation: Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043
References:
[1]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

F. CakoniD. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Pairs, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[3]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inv. Prob., 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004.  Google Scholar

[4]

F. CakoniD. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[5]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[6]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), vol. 60 of Math. Sci. Res. Inst. Publ., Cambridge University Press, Cambridge, 2013,527-580.  Google Scholar

[7]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.  doi: 10.1088/0266-5611/21/1/023.  Google Scholar

[8]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inv. Prob., 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[10]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math., 45 (1985), 1039-1053.  doi: 10.1137/0145064.  Google Scholar

[11]

D. Colton, P. Monk and J. Sun, Analytical and computational methods for transmission eigenvalues, Inv. Prob., 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011.  Google Scholar

[12]

D. ColtonL. Päivärinta and J. Sylvester, The interior transmission problem, Inv. Prob. Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[13]

P. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836.  Google Scholar

[14]

G. C. HsiaoF. LiuJ. Sun and L. Xu, A coupled BEM and FEM for the interior transmission problem in acoustics, J. Comput. Appl. Math., 235 (2011), 5213-5221.  doi: 10.1016/j.cam.2011.05.011.  Google Scholar

[15]

T.-M. HuangW.-Q. Huang and W.-W. Lin, A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems, SIAM J. Sci. Comput., 37 (2015), A2403-A2423.  doi: 10.1137/15M1018927.  Google Scholar

[16]

T.-M. HwangW.-W. LinW.-C. Wang and W. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys., 196 (2004), 208-232.  doi: 10.1016/j.jcp.2003.10.026.  Google Scholar

[17]

X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Trans. Math. Software, 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137.  Google Scholar

[18]

X. JiJ. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9.  Google Scholar

[19]

M. Kilmer and D. O'leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), 1204-1221.  doi: 10.1137/S0895479899345960.  Google Scholar

[20]

A. Kirsch, On the existence of transmission eigenvalues, Inv. Prob. Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.  Google Scholar

[21]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008.  Google Scholar

[22]

A. Kleefeld, Numerical Methods for Acoustic and Electromagnetic Scattering: Transmission Boundary-Value Problems, Interior Transmission Eigenvalues, and the Factorization Method, Habilitation Thesis, 2015. Google Scholar

[23]

T. Li, T. M. Huang, W. W. Lin and J. N. Wang, An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues, Inv. Prob., 33 (2017), 035009, 21pp. doi: 10.1088/1361-6420/aa5475.  Google Scholar

[24]

T. LiW.-Q. HuangW.-W. Lin and J. Liu, On spectral analysis and a novel algorithm for transmission eigenvalue problems, J. Sci. Comput., 64 (2015), 83-108.  doi: 10.1007/s10915-014-9923-0.  Google Scholar

[25]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.  doi: 10.1137/070697525.  Google Scholar

[26]

A. Serdyukov, I. Semchenko, S. Tretyakov and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications, Gordon and Breach Science, 2001. Google Scholar

[27]

J. Sun, Estimation of transmission eigenvalues and the index of refraction from cauchy data, Inv. Prob., 27 (2010), 015009, 11pp. doi: 10.1088/0266-5611/27/1/015009.  Google Scholar

[28]

J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874.  doi: 10.1137/100785478.  Google Scholar

[29]

J. Sun, An eigenvalue method using multiple frequency data for inverse scattering problems, Inv. Prob., 28(2) (2012), 025012, 15pp. doi: 10.1088/0266-5611/28/2/025012.  Google Scholar

[30]

J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton: CRC Press, 2017.  Google Scholar

[31]

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems, Winston Sons, Washington, D. C., 1977.  Google Scholar

show all references

References:
[1]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

F. CakoniD. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Pairs, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[3]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inv. Prob., 26 (2010), 074004, 14pp. doi: 10.1088/0266-5611/26/7/074004.  Google Scholar

[4]

F. CakoniD. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[5]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[6]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out II (ed. G. Uhlmann), vol. 60 of Math. Sci. Res. Inst. Publ., Cambridge University Press, Cambridge, 2013,527-580.  Google Scholar

[7]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory, Inv. Prob., 21 (2005), 383-398.  doi: 10.1088/0266-5611/21/1/023.  Google Scholar

[8]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inv. Prob., 12 (1996), 383-393.  doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[10]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region, SIAM J. Appl. Math., 45 (1985), 1039-1053.  doi: 10.1137/0145064.  Google Scholar

[11]

D. Colton, P. Monk and J. Sun, Analytical and computational methods for transmission eigenvalues, Inv. Prob., 26 (2010), 045011, 16pp. doi: 10.1088/0266-5611/26/4/045011.  Google Scholar

[12]

D. ColtonL. Päivärinta and J. Sylvester, The interior transmission problem, Inv. Prob. Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[13]

P. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836.  Google Scholar

[14]

G. C. HsiaoF. LiuJ. Sun and L. Xu, A coupled BEM and FEM for the interior transmission problem in acoustics, J. Comput. Appl. Math., 235 (2011), 5213-5221.  doi: 10.1016/j.cam.2011.05.011.  Google Scholar

[15]

T.-M. HuangW.-Q. Huang and W.-W. Lin, A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems, SIAM J. Sci. Comput., 37 (2015), A2403-A2423.  doi: 10.1137/15M1018927.  Google Scholar

[16]

T.-M. HwangW.-W. LinW.-C. Wang and W. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys., 196 (2004), 208-232.  doi: 10.1016/j.jcp.2003.10.026.  Google Scholar

[17]

X. Ji, J. Sun and T. Turner, Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues, ACM Trans. Math. Software, 38 (2012), Art. 29, 8 pp. doi: 10.1145/2331130.2331137.  Google Scholar

[18]

X. JiJ. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems, J. Sci. Comput., 60 (2014), 276-294.  doi: 10.1007/s10915-013-9794-9.  Google Scholar

[19]

M. Kilmer and D. O'leary, Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22 (2001), 1204-1221.  doi: 10.1137/S0895479899345960.  Google Scholar

[20]

A. Kirsch, On the existence of transmission eigenvalues, Inv. Prob. Imaging, 3 (2009), 155-172.  doi: 10.3934/ipi.2009.3.155.  Google Scholar

[21]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008.  Google Scholar

[22]

A. Kleefeld, Numerical Methods for Acoustic and Electromagnetic Scattering: Transmission Boundary-Value Problems, Interior Transmission Eigenvalues, and the Factorization Method, Habilitation Thesis, 2015. Google Scholar

[23]

T. Li, T. M. Huang, W. W. Lin and J. N. Wang, An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues, Inv. Prob., 33 (2017), 035009, 21pp. doi: 10.1088/1361-6420/aa5475.  Google Scholar

[24]

T. LiW.-Q. HuangW.-W. Lin and J. Liu, On spectral analysis and a novel algorithm for transmission eigenvalue problems, J. Sci. Comput., 64 (2015), 83-108.  doi: 10.1007/s10915-014-9923-0.  Google Scholar

[25]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753.  doi: 10.1137/070697525.  Google Scholar

[26]

A. Serdyukov, I. Semchenko, S. Tretyakov and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications, Gordon and Breach Science, 2001. Google Scholar

[27]

J. Sun, Estimation of transmission eigenvalues and the index of refraction from cauchy data, Inv. Prob., 27 (2010), 015009, 11pp. doi: 10.1088/0266-5611/27/1/015009.  Google Scholar

[28]

J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal., 49 (2011), 1860-1874.  doi: 10.1137/100785478.  Google Scholar

[29]

J. Sun, An eigenvalue method using multiple frequency data for inverse scattering problems, Inv. Prob., 28(2) (2012), 025012, 15pp. doi: 10.1088/0266-5611/28/2/025012.  Google Scholar

[30]

J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Boca Raton: CRC Press, 2017.  Google Scholar

[31]

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems, Winston Sons, Washington, D. C., 1977.  Google Scholar

Figure 1.  The target $D$ is inside some domain $\Omega$ ($\Gamma: = \partial \Omega$) which itself is surrounded by a curve $C$. The scattered field $u^s$ is due to the scattering of the incident field $u^i$ having a point source at ${\bf x}_0\in C$
Figure 2.  Four model domains that represent the region $D$
Figure 3.  The eigenvalues $\lambda$ of the QEP (20) in the intervel $[-80,250]$ for the four domains in Figure 2 with $\varepsilon({\bf x}) = -100$. The arrows point to the first positive eigenvalue of the QEP corresponding to each domain
Figure 4.  Reconstruction results of four targets in 3d surface figures and in 2d contour figures with $\varepsilon({\bf x}) = -100$ and $k^*\in (0, \sqrt{\beta^*})$. The domains enclosed by red curves are the exact targets $D$. The scattered fields used have $3\%$ noise
Figure 5.  Reconstruction results of four targets in 2d contour figures with $\varepsilon({\bf x}) = -100$ and the different $k\in (0, \sqrt{\beta^*})$. The domains enclosed by red curves are the exact targets $D$. The scattered fields have $3\%$ noise
Figure 6.  Numerical reconstruction results of a disk when $k^2$ is a transmission eigenvalue. The scattered fields have $3\%$ noise
Table 1.  Stiffness and mass matrices with $\varepsilon({\bf x}) <0$ for ${\bf x} \in \bar{D }$
stiffness matrix for interior meshes$K = [\int_D \nabla \phi_{i}\cdot\nabla \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$
stiffness matrix for interior/boundary meshes$E = [\int_D \nabla \phi_{i}\cdot\nabla \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
mass matrices for interior meshes$M_{1} = [\int_D \phi_{i} \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$
$M_{\varepsilon} = [-\int_D\varepsilon \phi_{i}\phi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{n \times n}$
mass matrices for interior/boundary meshes$F_{1} = [\int_D \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
$F_{\varepsilon} = [-\int_D\varepsilon \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
mass matrices for boundary meshes$G_{1} = [\int_D \psi_{i} \psi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{m \times m}$
$G_{\varepsilon} = [-\int_D\varepsilon \psi_{i} \psi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{m \times m}$
stiffness matrix for interior meshes$K = [\int_D \nabla \phi_{i}\cdot\nabla \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$
stiffness matrix for interior/boundary meshes$E = [\int_D \nabla \phi_{i}\cdot\nabla \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
mass matrices for interior meshes$M_{1} = [\int_D \phi_{i} \phi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{n \times n}$
$M_{\varepsilon} = [-\int_D\varepsilon \phi_{i}\phi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{n \times n}$
mass matrices for interior/boundary meshes$F_{1} = [\int_D \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
$F_{\varepsilon} = [-\int_D\varepsilon \phi_{i} \psi_{j}d{\bf x}] \in \mathbb{R}^{n \times m}$
mass matrices for boundary meshes$G_{1} = [\int_D \psi_{i} \psi_{j}d{\bf x}] \succ 0 \in \mathbb{R}^{m \times m}$
$G_{\varepsilon} = [-\int_D\varepsilon \psi_{i} \psi_{j}d{\bf x}]\succ 0 \in \mathbb{R}^{m \times m}$
Table 2.  Dimensions $n$, $m$ ($K \in \mathbb{R}^{n\times n}$, $E \in \mathbb{R}^{n \times m}$) of matrices for the benchmark problems with the mesh size $h \approx 0.004$
DomainDiskEllipsePeanutHeart
$(n, m)$(124631, 1150)(71546,976)(149051, 1871)(168548, 1492)
DomainDiskEllipsePeanutHeart
$(n, m)$(124631, 1150)(71546,976)(149051, 1871)(168548, 1492)
[1]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[2]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[3]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[4]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[5]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[6]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[7]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[8]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[9]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[10]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[11]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[12]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[13]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[14]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[15]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[16]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[17]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[18]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[19]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[20]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (98)
  • HTML views (140)
  • Cited by (2)

[Back to Top]