American Institute of Mathematical Sciences

Advanced
March  2015, 35(3): 1009-1037. doi: 10.3934/dcds.2015.35.1009

On the quenching behaviour of a semilinear wave equation modelling MEMS technology

Nikos I. Kavallaris 1, , Andrew A. Lacey 2, , Christos V. Nikolopoulos 3, and  Dimitrios E. Tzanetis 4,

1. 

Department of Mathematics, School of Science and Engineering, University of Chester, Thornton Science Park, Pool Lane, Ince Chester CH2 4NU, United Kingdom
2. 

Maxwell Institute for Mathematical Sciences & Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, United Kingdom
3. 

Department of Mathematics, University of the Aegean, GR-832 00 Karlovassi, Samos, Greece
4. 

Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece

Received  March 2014 Revised  July 2014 Published  October 2014

In this work we study the semilinear wave equation of the form \[ u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2}, \] with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any $\lambda>0.$ Then we focus on the singular behaviour of the solution, which occurs through finite-time quenching, i.e. when $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, investigating both conditions for quenching and the quenching profile of $u.$ To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.
Keywords: Electrostatic MEMS, quenching of solution, formal asymptotic, similarity analysis., hyperbolic problems.
Mathematics Subject Classification: Primary: 35L81, 35J60; Secondary: 74H35, 74G55, 74K1.
Citation: Nikos I. Kavallaris, Andrew A. Lacey, Christos V. Nikolopoulos, Dimitrios E. Tzanetis. On the quenching behaviour of a semilinear wave equation modelling MEMS technology. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1009-1037. doi: 10.3934/dcds.2015.35.1009
References:
[1]

R. C. Batra, M. Porfiri and D. Spinello, Electromechanical model of electrically actuated narrow microbeams, Jour. Microelectromechanical Systems, 15 (2006), 1175-1189. doi: 10.1109/JMEMS.2006.880204.

[2]

P. Bizon, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201. doi: 10.1088/0951-7715/17/6/009.

[3]

P. Bizon, D. Maison and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity, 20 (2007), 2061-2074. doi: 10.1088/0951-7715/20/9/003.

[4]

C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry, Journal of Engineering Mathematics, 66 (2010), 217-236. doi: 10.1007/s10665-009-9343-6.

[5]

P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations, SIAM Journal of Mathematical Analysis, 12 (1981), 893-903. doi: 10.1137/0512075.

[6]

C. Y. Chan and K. K. Nip, On the blow-up of $|u_{t t}|$ at quenching for semilinear Euler-Poisson-Darboux equations, Comp. Appl. Mat., 14 (1995), 185-190.

[7]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS, Courant Lect. Notes Math. 20, AMS/CIMS, New York, 2010.

[8]

S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quarterly Appl. Math., 51 (1993), 713-729.

[9]

G. Flores, Dynamics of a damped wave equation arising from MEMS, SIAM J. Appl. Math., 74 (2014), 1025-1035. doi: 10.1137/130914759.

[10]

G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., 67 (2007), 434-446. doi: 10.1137/060648866.

[11]

V. A. Galaktionov and S. I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation, Quart. Appl. Math., 61 (2003), 583-600.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803.

[13]

N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS, Methods Appl. Anal., 15 (2008), 361-376. doi: 10.4310/MAA.2008.v15.n3.a8.

[14]

J. L. Griffin, S. W. Schlosser, G. R. Ganger and D. F. Nagle, Operating system management of MEMS-based storage devices, Proceedings of the 4th Symposium on Operating Systems Design and Implementation (OSDI), (2000), 227-242.

[15]

Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behaviour of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338. doi: 10.1137/040613391.

[16]

J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in micro-electro mechanical systems, Quarterly Appl. Math., 67 (2009), 725-734.

[17]

J.-S. Guo and N. I. Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control, Disc. Cont. Dyn. Systems A, 32 (2012), 1723-1746. doi: 10.3934/dcds.2012.32.1723.

[18]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135-1163. doi: 10.1137/09077117X.

[19]

K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, Nonl. Anal: Theory, Methods & Applications, 74 (2011), 298-316. doi: 10.1016/j.na.2010.08.045.

[20]

N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Diff. Eqns. Appl., 15 (2008), 363-385. doi: 10.1007/s00030-008-7081-5.

[21]

N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic nonlocal problem modelling MEMS technology, Rocky Moun. J. Math., 41 (2011), 505-534. doi: 10.1216/RMJ-2011-41-2-505.

[22]

A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43 (1983), 1350-1366. doi: 10.1137/0143090.

[23]

P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 207 (2013), 139-158. doi: 10.1007/s00205-012-0559-7.

[24]

P. Laurencot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions,, Proc. London Math. Soc., (). 

[25]

J. Lega, A. E. Lindsay and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries, Journal of Nonlinear Science, 23 (2013), 807-834. doi: 10.1007/s00332-013-9169-2.

[26]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl., 155 (1989), 243-260. doi: 10.1007/BF01765943.

[27]

H. A. Levine, The phenomenon of quenching: A survey, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984), North-Holland Math. Stud., North-Holland, Amsterdam, 110 (1985), 275-286. doi: 10.1016/S0304-0208(08)72720-8.

[28]

H. A. Levine and M. W. Smiley, Abstract wave equations with a singular nonlinear forcing term, J. Math. Anal. Appl., 103 (1984), 409-427. doi: 10.1016/0022-247X(84)90138-0.

[29]

C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS, J. Diff. Equations, 256 (2014), 503-530. doi: 10.1016/j.jde.2013.09.010.

[30]

C. Liang and K. Zhang, Global solution of the initial boundary value problem to a hyperbolic nonlocal MEMS equation, Computers & Mathematics with Applications, 67 (2014), 549-554. doi: 10.1016/j.camwa.2013.11.012.

[31]

A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor， SIAM J. Appl. Math., 72 (2012), 935-958. doi: 10.1137/110832550.

[32]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Analysis, 253 (2007), 43-121. doi: 10.1016/j.jfa.2007.03.007.

[33]

J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump, ASEN 5519 Fluid-Structures Interactions, 2004.

[34]

F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations, Arch. Mat., 45 (2009), 115-124.

[35]

J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366. doi: 10.1023/A:1012292311304.

[36]

J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.

[37]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2003.

[38]

R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions, SIAM Journal of Math. Analysis, 20 (1989), 1081-1094. doi: 10.1137/0520072.

[39]

H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II, Theory and Performance, Sens. Actuat. A, 45 (1994), 67-84.

[40]

J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device, Patent No.: US 7,286,155 B1, Date of Patent: Oct. 23, 2007.

[41]

M. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. doi: 10.1007/978-1-4419-6020-7.

show all references

References:
[1]

R. C. Batra, M. Porfiri and D. Spinello, Electromechanical model of electrically actuated narrow microbeams, Jour. Microelectromechanical Systems, 15 (2006), 1175-1189. doi: 10.1109/JMEMS.2006.880204.

[2]

P. Bizon, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201. doi: 10.1088/0951-7715/17/6/009.

[3]

P. Bizon, D. Maison and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity, 20 (2007), 2061-2074. doi: 10.1088/0951-7715/20/9/003.

[4]

C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry, Journal of Engineering Mathematics, 66 (2010), 217-236. doi: 10.1007/s10665-009-9343-6.

[5]

P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations, SIAM Journal of Mathematical Analysis, 12 (1981), 893-903. doi: 10.1137/0512075.

[6]

C. Y. Chan and K. K. Nip, On the blow-up of $|u_{t t}|$ at quenching for semilinear Euler-Poisson-Darboux equations, Comp. Appl. Mat., 14 (1995), 185-190.

[7]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS, Courant Lect. Notes Math. 20, AMS/CIMS, New York, 2010.

[8]

S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quarterly Appl. Math., 51 (1993), 713-729.

[9]

G. Flores, Dynamics of a damped wave equation arising from MEMS, SIAM J. Appl. Math., 74 (2014), 1025-1035. doi: 10.1137/130914759.

[10]

G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., 67 (2007), 434-446. doi: 10.1137/060648866.

[11]

V. A. Galaktionov and S. I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation, Quart. Appl. Math., 61 (2003), 583-600.

[12]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803.

[13]

N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS, Methods Appl. Anal., 15 (2008), 361-376. doi: 10.4310/MAA.2008.v15.n3.a8.

[14]

J. L. Griffin, S. W. Schlosser, G. R. Ganger and D. F. Nagle, Operating system management of MEMS-based storage devices, Proceedings of the 4th Symposium on Operating Systems Design and Implementation (OSDI), (2000), 227-242.

[15]

Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behaviour of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338. doi: 10.1137/040613391.

[16]

J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in micro-electro mechanical systems, Quarterly Appl. Math., 67 (2009), 725-734.

[17]

J.-S. Guo and N. I. Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control, Disc. Cont. Dyn. Systems A, 32 (2012), 1723-1746. doi: 10.3934/dcds.2012.32.1723.

[18]

Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135-1163. doi: 10.1137/09077117X.

[19]

K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, Nonl. Anal: Theory, Methods & Applications, 74 (2011), 298-316. doi: 10.1016/j.na.2010.08.045.

[20]

N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Diff. Eqns. Appl., 15 (2008), 363-385. doi: 10.1007/s00030-008-7081-5.

[21]

N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic nonlocal problem modelling MEMS technology, Rocky Moun. J. Math., 41 (2011), 505-534. doi: 10.1216/RMJ-2011-41-2-505.

[22]

A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43 (1983), 1350-1366. doi: 10.1137/0143090.

[23]

P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 207 (2013), 139-158. doi: 10.1007/s00205-012-0559-7.

[24]

P. Laurencot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions,, Proc. London Math. Soc., (). 

[25]

J. Lega, A. E. Lindsay and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries, Journal of Nonlinear Science, 23 (2013), 807-834. doi: 10.1007/s00332-013-9169-2.

[26]

H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl., 155 (1989), 243-260. doi: 10.1007/BF01765943.

[27]

H. A. Levine, The phenomenon of quenching: A survey, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984), North-Holland Math. Stud., North-Holland, Amsterdam, 110 (1985), 275-286. doi: 10.1016/S0304-0208(08)72720-8.

[28]

H. A. Levine and M. W. Smiley, Abstract wave equations with a singular nonlinear forcing term, J. Math. Anal. Appl., 103 (1984), 409-427. doi: 10.1016/0022-247X(84)90138-0.

[29]

C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS, J. Diff. Equations, 256 (2014), 503-530. doi: 10.1016/j.jde.2013.09.010.

[30]

C. Liang and K. Zhang, Global solution of the initial boundary value problem to a hyperbolic nonlocal MEMS equation, Computers & Mathematics with Applications, 67 (2014), 549-554. doi: 10.1016/j.camwa.2013.11.012.

[31]

A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor， SIAM J. Appl. Math., 72 (2012), 935-958. doi: 10.1137/110832550.

[32]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Analysis, 253 (2007), 43-121. doi: 10.1016/j.jfa.2007.03.007.

[33]

J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump, ASEN 5519 Fluid-Structures Interactions, 2004.

[34]

F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations, Arch. Mat., 45 (2009), 115-124.

[35]

J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366. doi: 10.1023/A:1012292311304.

[36]

J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.

[37]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2003.

[38]

R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions, SIAM Journal of Math. Analysis, 20 (1989), 1081-1094. doi: 10.1137/0520072.

[39]

H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II, Theory and Performance, Sens. Actuat. A, 45 (1994), 67-84.

[40]

J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device, Patent No.: US 7,286,155 B1, Date of Patent: Oct. 23, 2007.

[41]

M. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011. doi: 10.1007/978-1-4419-6020-7.

[1]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160
[2]

Jong-Shenq Guo, Nikos I. Kavallaris. On a nonlocal parabolic problem arising in electrostatic MEMS control. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1723-1746. doi: 10.3934/dcds.2012.32.1723
[3]

Laurence Cherfils, Alain Miranville, Shuiran Peng, Chuanju Xu. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1601-1621. doi: 10.3934/dcdss.2019109
[4]

Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107
[5]

S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks and Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577
[6]

Jong-Shenq Guo, Bo-Chih Huang. Hyperbolic quenching problem with damping in the micro-electro mechanical system device. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 419-434. doi: 10.3934/dcdsb.2014.19.419
[7]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833
[8]

Qi Wang, Yanyan Zhang. Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 797-816. doi: 10.3934/cpaa.2021199
[9]

Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
[10]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023
[11]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control and Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
[12]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799
[13]

M. Chipot, A. Rougirel. On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 319-338. doi: 10.3934/dcdsb.2001.1.319
[14]

Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial and Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689
[15]

Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[16]

Joachim Escher, Christina Lienstromberg. A survey on second order free boundary value problems modelling MEMS with general permittivity profile. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 745-771. doi: 10.3934/dcdss.2017038
[17]

Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627
[18]

Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis, Kostas Kouramas. Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 177-206. doi: 10.3934/naco.2019047
[19]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure and Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[20]

Sergio Frigeri. Asymptotic behavior of a hyperbolic system arising in ferroelectricity. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1393-1414. doi: 10.3934/cpaa.2008.7.1393

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy