2016, 9(2): 585-597. doi: 10.3934/dcdss.2016013

Cellular instabilities analyzed by multi-scale Fourier series: A review

1. 

LEM3, Laboratoire d'Etudes des Microstructures et de Mécanique des Matériaux, UMR CNRS 7239, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 01, France, France

2. 

Department of Mechanics and Engineering Science, Fudan University, 220 Handan Road, 200433 Shanghai, China

3. 

Laboratoire d'Ingénierie et Matériaux LIMAT, Faculté des Sciences Ben M'Sik, Université Hassan II de Casablanca, Sidi Othman, Casablanca, Morocco, Morocco, Morocco

4. 

School of Civil Engineering, Wuhan University, 8 South Road of East Lake, 430072 Wuhan, China, China

5. 

Université de Montpellier, Laboratoire de Mécanique et Génie Civil, UMR CNRS 5508, CC048 Place Eugène Bataillon, 34095 Montpellier Cedex 05, France

Received  April 2015 Revised  October 2015 Published  March 2016

The paper is concerned by multi-scale methods to describe instability pattern formation, especially the method of Fourier series with variable coefficients. In this respect, various numerical tools are available. For instance in the case of membrane models, shell finite element codes can predict the details of the wrinkles, but with difficulties due to the large number of unknowns and the existence of many solutions. Macroscopic models are also available, but they account only for the effect of wrinkling on membrane behavior. A Fourier-related method has been introduced in order to modelize the main features of the wrinkles, but by using partial differential equations only at a macroscopic level. Within this method, the solution is sought in the form of few terms of Fourier series whose coefficients vary more slowly than the oscillations. The recent progresses about this Fourier-related method are reviewed and discussed.
Citation: Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013
References:
[1]

S. Abdelkhalek, Un Exemple de Flambage Sous Contraintes Internes: Étude des Défauts de Planéité en Laminage à Froid Des Tôles Minces,, Doctoral dissertation, (2010).

[2]

S. Abdelkhalek, P. Montmitonnet, M. Potier-Ferry, H. Zahrouni, N. Legrand and P. Buessler, Strip flatness modelling including buckling phenomena during thin strip cold rolling,, Ironmaking and Steelmaking, 37 (2010), 290. doi: 10.1179/030192310X12646889255708.

[3]

J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on non-linear elastic foundations,, Internat. J. Solids Structures, 6 (1970), 1341.

[4]

K. Attipou, H. Hu, F. Mohri, M. Potier-Ferry and S. Belouettar, Thermal wrinkling of thin membranes using a Fourier-related double scale approach,, Thin-Walled Structures, 94 (2015), 532. doi: 10.1016/j.tws.2015.04.034.

[5]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North Holland Publ, (1978).

[6]

N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations,, Gordon and Breach, (1963).

[7]

N. Bowden, S. Brittain, A. G. Evans, J. W. Hutchinson and G. M. Whitesides, Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer,, Nature, 393 (1998), 146.

[8]

F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud and P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of non- linear oscillators,, Nature Physics, 7 (2011), 56.

[9]

M. C. Cross, P. G. Daniels, P. C. Hohenberg and E. D. Siggia, Phase-winding solutions in a finite container above the convective threshold,, J. Fluid Mech., 127 (1983), 155. doi: 10.1017/S0022112083002670.

[10]

M. C. Cross and P. C. Hohenberg, Pattern formation out of equilibrium,, Rev. Modern Phys., 65 (1993), 851.

[11]

N. Damil and M. Potier-Ferry, Amplitude equations for cellular instabilities,, Dynamics and Stability of Systems, 7 (1992), 1. doi: 10.1080/02681119208806124.

[12]

N. Damil and M. Potier-Ferry, A generalized continuum approach to describe instability pattern formation by a multiple scale analysis,, Comptes Rendus Mecanique, 334 (2006), 674. doi: 10.1016/j.crme.2006.09.002.

[13]

N. Damil and M. Potier-Ferry, A generalized continuum approach to predict local buckling patterns of thin structures,, European Journal of Computational Mechanics, 17 (2008), 945.

[14]

N. Damil and M. Potier-Ferry, Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients,, J. Mech. Phys. Solids, 58 (2010), 1139. doi: 10.1016/j.jmps.2010.04.002.

[15]

N. Damil, M. Potier-Ferry and H. Hu, New nonlinear multiscale models for membrane wrinkling,, Comptes Rendus Mecanique, 341 (2013), 616.

[16]

N. Damil, M. Potier-Ferry and H. Hu, Membrane wrinkling revisited from a multi-scale point of view,, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014). doi: 10.1186/2213-7467-1-6.

[17]

A. Eriksson and A. Nordmark, Instability of hyper-elastic balloon-shaped space membranes under pressure loads,, Comput. Methods Appl. Mech. Engrg., 237 (2012), 118. doi: 10.1016/j.cma.2012.05.012.

[18]

F. Feyel and J. L. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials,, Comput. Methods Appl. Mech. Engrg., 183 (2000), 309. doi: 10.1016/S0045-7825(99)00224-8.

[19]

G. Geymonat, S. Muller and N. Triantafyllidis, Homogenization of nonlinearly elastic materials: Microscopic bifurcation and macroscopic loss of rank-one convexity,, Arch. Ration. Mech. Anal., 122 (1993), 231. doi: 10.1007/BF00380256.

[20]

R. Hoyle, Pattern Formation, An Introduction to Methods,, Cambrige University Press, (2006). doi: 10.1017/CBO9780511616051.

[21]

H. Hu, N. Damil and M. Potier-Ferry, A bridging technique to analyze the influence of boundary conditions on instability patterns,, J. Comput. Phys., 230 (2011), 3753. doi: 10.1016/j.jcp.2011.01.044.

[22]

Q. Huang, H. Hu, K. Yu, M. Potier-Ferry, N. Damil and S. Belouettar, Macroscopic simulation of membrane wrinkling for various loading cases,, Internat. J. Solids Structures, 64-65 (2015), 64. doi: 10.1016/j.ijsolstr.2015.04.003.

[23]

G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord, Cellular buckling in long structures,, Nonlinear Dynamics, 21 (2000), 3. doi: 10.1023/A:1008398006403.

[24]

G. Iooss, A. Mielke and Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems,, Eur. J. Mech. B Fluids, 8 (1989), 229.

[25]

Y. Lecieux and R. Bouzidi, Experimentation analysis on membrane wrinkling under biaxial load - Comparison with bifurcation analysis,, Internat. J. Solids Structures, 47 (2010), 2459.

[26]

B. Li, Y. P. Cao, X. Q. Feng and H. J. Gao, Mechanics of morphological instabilities and surface wrinkling in soft materials: A review,, Soft Matter, 8 (2012), 5728. doi: 10.1039/c2sm00011c.

[27]

Y. Liu, K. Yu, H. Hu, S. Belouettar and M. Potier-Ferry, A Fourier-related double scale analysis on instability phenomena of sandwich beams,, Internat. J. Solids Structures, 49 (2012), 3077.

[28]

K. Mhada, B. Braikat and N. Damil, A 2D Fourier double scale analysis of global-local instability interaction in sandwich structures,, 21ème Congrès Français de Mécanique, (2013).

[29]

K. Mhada, B. Braikat, H. Hu, N. Damil and M. Potier-Ferry, About macroscopic models of instability pattern formation,, Internat. J. Solids Structures, 49 (2012), 2978. doi: 10.1016/j.ijsolstr.2012.05.033.

[30]

H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure,, Comput. Methods Appl. Mech. Engrg., 157 (1998), 69. doi: 10.1016/S0045-7825(97)00218-1.

[31]

R. Nakhoul, P. Montmitonnet and M. Potier-Ferry, Multi-scale method modeling of thin sheet buckling under residual stresses in the context of strip rolling,, Internat. J. Solids Structures, 66 (2015), 62. doi: 10.1016/j.ijsolstr.2015.03.028.

[32]

A. H. Nayfeh, Perturbation Methods,, John Wiley and Sons, (1973).

[33]

A. C. Newell and J. A. Whitehead, Finite band width, finite amplitude convection,, J. Fluid Mech., 38 (1969), 279. doi: 10.1017/S0022112069000176.

[34]

S. Nezamabadi, J. Yvonnet, H. Zahrouni and M. Potier-Ferry, A multilevel computational strategy for handling microscopic and macroscopic instabilities,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2099. doi: 10.1016/j.cma.2009.02.026.

[35]

Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures,, Journal de Physique, 42 (1981), 515. doi: 10.1051/jphys:01981004204051500.

[36]

R. Rossi, M. Lazzari, R. Vitaliani and E. Onate, Simulation of light-weight membrane structures by wrinkling model,, Internat. J. Numer. Methods Engrg, 62 (2005), 2127. doi: 10.1002/nme.1266.

[37]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980).

[38]

L. A. Segel, Distant side walls cause slow amplitude modulation of cellular convection,, J. Fluid Mech., 38 (1969), 203. doi: 10.1017/S0022112069000127.

[39]

R. J. M. Smit, W. A. M. Brekelmans and H. E. H. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling,, Comput. Methods Appl. Mech. Engrg., 155 (1998), 181. doi: 10.1016/S0045-7825(97)00139-4.

[40]

P. Suquet, Plasticité et Homogénéisation,, Doctoral dissertation, (1982).

[41]

M. A. Wadee and M. Farsi, Cellular buckling in stiffened plates,, Proc. R. Soc. A, 470 (2014). doi: 10.1098/rspa.2014.0094.

[42]

J. E Wesfreid and S. Zaleski editors, Cellular Structures in Instabilities,, Lecture Notes in Physics, (1984). doi: 10.1007/3-540-13879-X.

[43]

Y. W. Wong and S. Pellegrino, Wrinkled membranes-Part1: Experiments,, Journal of Mechanics of Materials and Structures, 1 (2006), 3.

[44]

F. Xu, H. Hu, M. Potier-Ferry and S. Belouettar, Bridging techniques in a multi-scale modeling of pattern formation,, Internat. J. Solids Structures, 51 (2014), 3119. doi: 10.1016/j.ijsolstr.2014.05.011.

[45]

F. Xu and M. Potier-Ferry, A multi-scale modeling framework for instabilities of film/substrate systems,, J. Mech. Phys. Solids, 86 (2016), 150. doi: 10.1016/j.jmps.2015.10.003.

[46]

K. Yu, H. Hu, S. Chen, S. Belouettar and M. Potier-Ferry, Multi-scale techniques to analyze instabilities in sandwich structures,, Composite Structures, 96 (2013), 751. doi: 10.1016/j.compstruct.2012.10.007.

show all references

References:
[1]

S. Abdelkhalek, Un Exemple de Flambage Sous Contraintes Internes: Étude des Défauts de Planéité en Laminage à Froid Des Tôles Minces,, Doctoral dissertation, (2010).

[2]

S. Abdelkhalek, P. Montmitonnet, M. Potier-Ferry, H. Zahrouni, N. Legrand and P. Buessler, Strip flatness modelling including buckling phenomena during thin strip cold rolling,, Ironmaking and Steelmaking, 37 (2010), 290. doi: 10.1179/030192310X12646889255708.

[3]

J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on non-linear elastic foundations,, Internat. J. Solids Structures, 6 (1970), 1341.

[4]

K. Attipou, H. Hu, F. Mohri, M. Potier-Ferry and S. Belouettar, Thermal wrinkling of thin membranes using a Fourier-related double scale approach,, Thin-Walled Structures, 94 (2015), 532. doi: 10.1016/j.tws.2015.04.034.

[5]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North Holland Publ, (1978).

[6]

N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations,, Gordon and Breach, (1963).

[7]

N. Bowden, S. Brittain, A. G. Evans, J. W. Hutchinson and G. M. Whitesides, Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer,, Nature, 393 (1998), 146.

[8]

F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud and P. Damman, Multiple-length-scale elastic instability mimics parametric resonance of non- linear oscillators,, Nature Physics, 7 (2011), 56.

[9]

M. C. Cross, P. G. Daniels, P. C. Hohenberg and E. D. Siggia, Phase-winding solutions in a finite container above the convective threshold,, J. Fluid Mech., 127 (1983), 155. doi: 10.1017/S0022112083002670.

[10]

M. C. Cross and P. C. Hohenberg, Pattern formation out of equilibrium,, Rev. Modern Phys., 65 (1993), 851.

[11]

N. Damil and M. Potier-Ferry, Amplitude equations for cellular instabilities,, Dynamics and Stability of Systems, 7 (1992), 1. doi: 10.1080/02681119208806124.

[12]

N. Damil and M. Potier-Ferry, A generalized continuum approach to describe instability pattern formation by a multiple scale analysis,, Comptes Rendus Mecanique, 334 (2006), 674. doi: 10.1016/j.crme.2006.09.002.

[13]

N. Damil and M. Potier-Ferry, A generalized continuum approach to predict local buckling patterns of thin structures,, European Journal of Computational Mechanics, 17 (2008), 945.

[14]

N. Damil and M. Potier-Ferry, Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients,, J. Mech. Phys. Solids, 58 (2010), 1139. doi: 10.1016/j.jmps.2010.04.002.

[15]

N. Damil, M. Potier-Ferry and H. Hu, New nonlinear multiscale models for membrane wrinkling,, Comptes Rendus Mecanique, 341 (2013), 616.

[16]

N. Damil, M. Potier-Ferry and H. Hu, Membrane wrinkling revisited from a multi-scale point of view,, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014). doi: 10.1186/2213-7467-1-6.

[17]

A. Eriksson and A. Nordmark, Instability of hyper-elastic balloon-shaped space membranes under pressure loads,, Comput. Methods Appl. Mech. Engrg., 237 (2012), 118. doi: 10.1016/j.cma.2012.05.012.

[18]

F. Feyel and J. L. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials,, Comput. Methods Appl. Mech. Engrg., 183 (2000), 309. doi: 10.1016/S0045-7825(99)00224-8.

[19]

G. Geymonat, S. Muller and N. Triantafyllidis, Homogenization of nonlinearly elastic materials: Microscopic bifurcation and macroscopic loss of rank-one convexity,, Arch. Ration. Mech. Anal., 122 (1993), 231. doi: 10.1007/BF00380256.

[20]

R. Hoyle, Pattern Formation, An Introduction to Methods,, Cambrige University Press, (2006). doi: 10.1017/CBO9780511616051.

[21]

H. Hu, N. Damil and M. Potier-Ferry, A bridging technique to analyze the influence of boundary conditions on instability patterns,, J. Comput. Phys., 230 (2011), 3753. doi: 10.1016/j.jcp.2011.01.044.

[22]

Q. Huang, H. Hu, K. Yu, M. Potier-Ferry, N. Damil and S. Belouettar, Macroscopic simulation of membrane wrinkling for various loading cases,, Internat. J. Solids Structures, 64-65 (2015), 64. doi: 10.1016/j.ijsolstr.2015.04.003.

[23]

G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord, Cellular buckling in long structures,, Nonlinear Dynamics, 21 (2000), 3. doi: 10.1023/A:1008398006403.

[24]

G. Iooss, A. Mielke and Y. Demay, Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems,, Eur. J. Mech. B Fluids, 8 (1989), 229.

[25]

Y. Lecieux and R. Bouzidi, Experimentation analysis on membrane wrinkling under biaxial load - Comparison with bifurcation analysis,, Internat. J. Solids Structures, 47 (2010), 2459.

[26]

B. Li, Y. P. Cao, X. Q. Feng and H. J. Gao, Mechanics of morphological instabilities and surface wrinkling in soft materials: A review,, Soft Matter, 8 (2012), 5728. doi: 10.1039/c2sm00011c.

[27]

Y. Liu, K. Yu, H. Hu, S. Belouettar and M. Potier-Ferry, A Fourier-related double scale analysis on instability phenomena of sandwich beams,, Internat. J. Solids Structures, 49 (2012), 3077.

[28]

K. Mhada, B. Braikat and N. Damil, A 2D Fourier double scale analysis of global-local instability interaction in sandwich structures,, 21ème Congrès Français de Mécanique, (2013).

[29]

K. Mhada, B. Braikat, H. Hu, N. Damil and M. Potier-Ferry, About macroscopic models of instability pattern formation,, Internat. J. Solids Structures, 49 (2012), 2978. doi: 10.1016/j.ijsolstr.2012.05.033.

[30]

H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure,, Comput. Methods Appl. Mech. Engrg., 157 (1998), 69. doi: 10.1016/S0045-7825(97)00218-1.

[31]

R. Nakhoul, P. Montmitonnet and M. Potier-Ferry, Multi-scale method modeling of thin sheet buckling under residual stresses in the context of strip rolling,, Internat. J. Solids Structures, 66 (2015), 62. doi: 10.1016/j.ijsolstr.2015.03.028.

[32]

A. H. Nayfeh, Perturbation Methods,, John Wiley and Sons, (1973).

[33]

A. C. Newell and J. A. Whitehead, Finite band width, finite amplitude convection,, J. Fluid Mech., 38 (1969), 279. doi: 10.1017/S0022112069000176.

[34]

S. Nezamabadi, J. Yvonnet, H. Zahrouni and M. Potier-Ferry, A multilevel computational strategy for handling microscopic and macroscopic instabilities,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2099. doi: 10.1016/j.cma.2009.02.026.

[35]

Y. Pomeau and S. Zaleski, Wavelength selection in one-dimensional cellular structures,, Journal de Physique, 42 (1981), 515. doi: 10.1051/jphys:01981004204051500.

[36]

R. Rossi, M. Lazzari, R. Vitaliani and E. Onate, Simulation of light-weight membrane structures by wrinkling model,, Internat. J. Numer. Methods Engrg, 62 (2005), 2127. doi: 10.1002/nme.1266.

[37]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980).

[38]

L. A. Segel, Distant side walls cause slow amplitude modulation of cellular convection,, J. Fluid Mech., 38 (1969), 203. doi: 10.1017/S0022112069000127.

[39]

R. J. M. Smit, W. A. M. Brekelmans and H. E. H. Meijer, Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling,, Comput. Methods Appl. Mech. Engrg., 155 (1998), 181. doi: 10.1016/S0045-7825(97)00139-4.

[40]

P. Suquet, Plasticité et Homogénéisation,, Doctoral dissertation, (1982).

[41]

M. A. Wadee and M. Farsi, Cellular buckling in stiffened plates,, Proc. R. Soc. A, 470 (2014). doi: 10.1098/rspa.2014.0094.

[42]

J. E Wesfreid and S. Zaleski editors, Cellular Structures in Instabilities,, Lecture Notes in Physics, (1984). doi: 10.1007/3-540-13879-X.

[43]

Y. W. Wong and S. Pellegrino, Wrinkled membranes-Part1: Experiments,, Journal of Mechanics of Materials and Structures, 1 (2006), 3.

[44]

F. Xu, H. Hu, M. Potier-Ferry and S. Belouettar, Bridging techniques in a multi-scale modeling of pattern formation,, Internat. J. Solids Structures, 51 (2014), 3119. doi: 10.1016/j.ijsolstr.2014.05.011.

[45]

F. Xu and M. Potier-Ferry, A multi-scale modeling framework for instabilities of film/substrate systems,, J. Mech. Phys. Solids, 86 (2016), 150. doi: 10.1016/j.jmps.2015.10.003.

[46]

K. Yu, H. Hu, S. Chen, S. Belouettar and M. Potier-Ferry, Multi-scale techniques to analyze instabilities in sandwich structures,, Composite Structures, 96 (2013), 751. doi: 10.1016/j.compstruct.2012.10.007.

[1]

Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451

[2]

Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Multi-scale model of bladder cancer development. Conference Publications, 2011, 2011 (Special) : 803-812. doi: 10.3934/proc.2011.2011.803

[3]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449

[4]

Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau. Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects. Conference Publications, 2013, 2013 (special) : 69-76. doi: 10.3934/proc.2013.2013.69

[5]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[6]

Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062

[7]

Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200

[8]

Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021

[9]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020

[10]

Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial & Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819

[11]

Fágner D. Araruna, Flank D. M. Bezerra, Milton L. Oliveira. Rate of attraction for a semilinear thermoelastic system with variable coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3211-3226. doi: 10.3934/dcdsb.2018316

[12]

Abderrahman Iggidr, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Multi-compartment models. Conference Publications, 2007, 2007 (Special) : 506-519. doi: 10.3934/proc.2007.2007.506

[13]

Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic & Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031

[14]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349

[15]

M. Eller, Roberto Triggiani. Exact/approximate controllability of thermoelastic plates with variable thermal coefficients. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 283-302. doi: 10.3934/dcds.2001.7.283

[16]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609

[17]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032

[18]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129

[19]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445

[20]

Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151

[Back to Top]