American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020341

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Xiao Wu 1, and  Mingkang Ni 2,,

1. 

School of Mathematical Sciences, East China Normal University, Shanghai 200000, P. R. China
2. 

School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200000, P. R. China

* Corresponding author: Mingkang Ni

Received  June 2019 Revised  October 2019 Published  April 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China (No. 11871217) and the Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

Keywords: discontinuous reactive term, periodic solution, asymptotic approximation, lower and upper solutions, small parameter.
Mathematics Subject Classification: Primary: 34D15, 34E15; Secondary: 34K13.
Citation: Xiao Wu, Mingkang Ni. Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020341
References:
[1]

F. M. Arscott, Periodic-parabolic boundary value problems and positivity, Bulletin of the London Mathematical Society, 24 (1991). doi: 10.1112/blms/24.6.619.  Google Scholar

[2]

C. De Coster, F. Obersnel and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, Handbook of Differential Equations: Ordinary Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 203-339.  doi: 10.1016/S1874-5725(06)80007-6.  Google Scholar

[3]

Z. J. Du and Z. S. Feng, Existence and asymptotic behaviors of traveling waves of a modified vector-disease model, Commun. Pure Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.  Google Scholar

[4]

Z. J. Du, J. Li and X. W. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

G. Hek, Geometric singular perturbation theory in biological pratice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[6]

N. T. Levashova, O. A. Nikolaeva and A. D. Pashkin, Simulation of the temperature distribution at the water-air interface using the theory of contrast structures, Vestnik Moskov. Univ. Ser. III Fiz. Astronom., (2015), 12–16.  Google Scholar

[7]

N. T. Levashova, Y. V. Mukhartova and W. A. Davydova, Application of the theory of constract structures to the description of the wind velocity field in the space-inhomogeneous vegetable cover, Moscow University Physics Bulletin, 3 (2015), 3-10.   Google Scholar

[8]

N. T. Levashova and O. A. Nikolaeva, The heat equation solution near the interface between two media, Lomonosov Moscow State University, 24 (2017), 339-352.  doi: 10.18255/1818-1015-2017-3-339-352.  Google Scholar

[9]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Time-independent reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 57 (2017), 854-866.  doi: 10.1134/S0965542517050062.  Google Scholar

[10]

N. T. Levashova, N. N. Nefedov, O. A. Nikolaeva, A. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms, Mathematical Methods in the Applied Sciences, 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

[11]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Asymptotic stability of a stationary solution of a multidimensional reaction-diffusion equation with a discontinuous source, Comput. Math. Math. Phys., 59 (2019), 573-582.  doi: 10.1134/S0965542519040109.  Google Scholar

[12]

E. A. Mikhailov, Wavefronts of the magnetic field in galaxies: Asymptotic and numerical approaches, Magnetohydrodynamics, 52 (2016), 117-125.   Google Scholar

[13]

N. N. Nefedov and M. K. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[14]

N. N. Nefedov, The method of differential inequalities for some singularly perturbed partial differential equations, Differential Equations, 31 (1995), 668-671.   Google Scholar

[15]

N. N. Nefedov and M. A. Davydova, Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems, Differential Equations, 48 (2012), 745-755.  doi: 10.1134/S0012266112050138.  Google Scholar

[16]

N. N. Nefedov and M. A. Davydova, Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations, Differential Equations, 49 (2013), 688-706.  doi: 10.1134/S0012266113060049.  Google Scholar

[17]

N. N. Nefedov, An asymptotic method of differential inequalities for the investigation of periodic contrast structures: Existence, asymptotics and stability, Differential Equations, 36 (2000), 298-305.  doi: 10.1007/BF02754216.  Google Scholar

[18]

A. Orlov, N. Levashova and T. Burbaev, The use of asymptotic methods for modelling of the carriers wave functions in the Si/SiGe heterostructures with quantum-confined layers, Journal of Physics: Conference Series, 586 (2014). doi: 10.1088/1742-6596/586/1/012003.  Google Scholar

[19]

A. O. Orlov, N. T. Levashova and N. N. Nefedov, Solution of contrast structure type for a parabolic reaction-diffusion problem in a medium with discontinuous characteristics, Differential Equations, 54 (2018), 669-686.  doi: 10.1134/S0012266118050105.  Google Scholar

[20]

C. V. Pao, Periodic solutions of parabolic systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 234 (1999), 695-716.  doi: 10.1006/jmaa.1999.6412.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[22]

S. I. Pohozaev, On equations of the form $ \Delta u = f(x, u, Du)$, Mathematics of the USSR-Sbornik, 41 (1982), 269-280.  doi: 10.1070/SM1982v041n02ABEH002233.  Google Scholar

[23]

V. N. Pavlenko and O. V. Ul'Yanova, The method of upper and lower solutions for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat., 42 (1998), 69-76.   Google Scholar

[24]

V. N. Pavlenko, Strong solutions of periodic parabolic problems with discontinuous nonlinearities, Differential Equations, 52 (2016), 505-516.  doi: 10.1134/S0012266116040108.  Google Scholar

[25]

Y. Pang, M. K. Ni, N. T. Levashova and O. A. Nikolaeva, Internal layers for a singualrly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[26]

Y. F. Pang, M. K. Ni and M. A. Davaydova, Contrast structures in problems for a stationary equation of reaction-diffusion-advection type with discontinuous nonlinearity, Mathematical Notes, 104 (2018), 735-744.  doi: 10.4213/mzm11699.  Google Scholar

[27]

Y. F. Pang, M. K. Ni and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right-hand side, Differential Equations, 54 (2018), 1583-1594.  doi: 10.1134/S0012266118120054.  Google Scholar

[28]

X. H. Shang and Z. J. Du, Existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, Math. Methods Appl. Sci., 39 (2016), 3035-3042.  doi: 10.1002/mma.3750.  Google Scholar

[29]

A. B. Vasil'Yeva, Step-Like Contrasting Structures for a System of Singularly Perturbed Equations, 1994. Google Scholar

[30]

A. B. Vasil'Yeva and V. F. Butuzov, Asymptitic Expansions of Solutions to Singularly Perturbed Equations, 1973. Google Scholar

[31]

A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 268 (2010), 258-273.  doi: 10.1134/S0081543810010189.  Google Scholar

[32]

V. T. Volkov and N. N. Nefëdov, Development of the asymptotic method of differential inequalities for investigation of periodic constrast structures in reaction-diffusion equations, Computational Mathematics and Mathematical Physics, 46 (2006), 585-593.  doi: 10.1134/s0965542506040075.  Google Scholar

[33]

V. T. Volkov and N. N. Nefëdov, Periodic solutions with boundary layers of a singularly perturbed reaction-diffusion model, Computational Mathematics and Mathematical Physics, 34 (1994), 1133-1140.   Google Scholar

[34]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.  Google Scholar

[35]

Y. Xu, Z. J. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized burgers-kdv equation, Nolinear Dynamics, 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

show all references

References:
[1]

F. M. Arscott, Periodic-parabolic boundary value problems and positivity, Bulletin of the London Mathematical Society, 24 (1991). doi: 10.1112/blms/24.6.619.  Google Scholar

[2]

C. De Coster, F. Obersnel and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, Handbook of Differential Equations: Ordinary Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 203-339.  doi: 10.1016/S1874-5725(06)80007-6.  Google Scholar

[3]

Z. J. Du and Z. S. Feng, Existence and asymptotic behaviors of traveling waves of a modified vector-disease model, Commun. Pure Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.  Google Scholar

[4]

Z. J. Du, J. Li and X. W. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

G. Hek, Geometric singular perturbation theory in biological pratice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[6]

N. T. Levashova, O. A. Nikolaeva and A. D. Pashkin, Simulation of the temperature distribution at the water-air interface using the theory of contrast structures, Vestnik Moskov. Univ. Ser. III Fiz. Astronom., (2015), 12–16.  Google Scholar

[7]

N. T. Levashova, Y. V. Mukhartova and W. A. Davydova, Application of the theory of constract structures to the description of the wind velocity field in the space-inhomogeneous vegetable cover, Moscow University Physics Bulletin, 3 (2015), 3-10.   Google Scholar

[8]

N. T. Levashova and O. A. Nikolaeva, The heat equation solution near the interface between two media, Lomonosov Moscow State University, 24 (2017), 339-352.  doi: 10.18255/1818-1015-2017-3-339-352.  Google Scholar

[9]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Time-independent reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 57 (2017), 854-866.  doi: 10.1134/S0965542517050062.  Google Scholar

[10]

N. T. Levashova, N. N. Nefedov, O. A. Nikolaeva, A. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms, Mathematical Methods in the Applied Sciences, 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

[11]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Asymptotic stability of a stationary solution of a multidimensional reaction-diffusion equation with a discontinuous source, Comput. Math. Math. Phys., 59 (2019), 573-582.  doi: 10.1134/S0965542519040109.  Google Scholar

[12]

E. A. Mikhailov, Wavefronts of the magnetic field in galaxies: Asymptotic and numerical approaches, Magnetohydrodynamics, 52 (2016), 117-125.   Google Scholar

[13]

N. N. Nefedov and M. K. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[14]

N. N. Nefedov, The method of differential inequalities for some singularly perturbed partial differential equations, Differential Equations, 31 (1995), 668-671.   Google Scholar

[15]

N. N. Nefedov and M. A. Davydova, Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems, Differential Equations, 48 (2012), 745-755.  doi: 10.1134/S0012266112050138.  Google Scholar

[16]

N. N. Nefedov and M. A. Davydova, Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations, Differential Equations, 49 (2013), 688-706.  doi: 10.1134/S0012266113060049.  Google Scholar

[17]

N. N. Nefedov, An asymptotic method of differential inequalities for the investigation of periodic contrast structures: Existence, asymptotics and stability, Differential Equations, 36 (2000), 298-305.  doi: 10.1007/BF02754216.  Google Scholar

[18]

A. Orlov, N. Levashova and T. Burbaev, The use of asymptotic methods for modelling of the carriers wave functions in the Si/SiGe heterostructures with quantum-confined layers, Journal of Physics: Conference Series, 586 (2014). doi: 10.1088/1742-6596/586/1/012003.  Google Scholar

[19]

A. O. Orlov, N. T. Levashova and N. N. Nefedov, Solution of contrast structure type for a parabolic reaction-diffusion problem in a medium with discontinuous characteristics, Differential Equations, 54 (2018), 669-686.  doi: 10.1134/S0012266118050105.  Google Scholar

[20]

C. V. Pao, Periodic solutions of parabolic systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 234 (1999), 695-716.  doi: 10.1006/jmaa.1999.6412.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[22]

S. I. Pohozaev, On equations of the form $ \Delta u = f(x, u, Du)$, Mathematics of the USSR-Sbornik, 41 (1982), 269-280.  doi: 10.1070/SM1982v041n02ABEH002233.  Google Scholar

[23]

V. N. Pavlenko and O. V. Ul'Yanova, The method of upper and lower solutions for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat., 42 (1998), 69-76.   Google Scholar

[24]

V. N. Pavlenko, Strong solutions of periodic parabolic problems with discontinuous nonlinearities, Differential Equations, 52 (2016), 505-516.  doi: 10.1134/S0012266116040108.  Google Scholar

[25]

Y. Pang, M. K. Ni, N. T. Levashova and O. A. Nikolaeva, Internal layers for a singualrly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[26]

Y. F. Pang, M. K. Ni and M. A. Davaydova, Contrast structures in problems for a stationary equation of reaction-diffusion-advection type with discontinuous nonlinearity, Mathematical Notes, 104 (2018), 735-744.  doi: 10.4213/mzm11699.  Google Scholar

[27]

Y. F. Pang, M. K. Ni and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right-hand side, Differential Equations, 54 (2018), 1583-1594.  doi: 10.1134/S0012266118120054.  Google Scholar

[28]

X. H. Shang and Z. J. Du, Existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, Math. Methods Appl. Sci., 39 (2016), 3035-3042.  doi: 10.1002/mma.3750.  Google Scholar

[29]

A. B. Vasil'Yeva, Step-Like Contrasting Structures for a System of Singularly Perturbed Equations, 1994. Google Scholar

[30]

A. B. Vasil'Yeva and V. F. Butuzov, Asymptitic Expansions of Solutions to Singularly Perturbed Equations, 1973. Google Scholar

[31]

A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 268 (2010), 258-273.  doi: 10.1134/S0081543810010189.  Google Scholar

[32]

V. T. Volkov and N. N. Nefëdov, Development of the asymptotic method of differential inequalities for investigation of periodic constrast structures in reaction-diffusion equations, Computational Mathematics and Mathematical Physics, 46 (2006), 585-593.  doi: 10.1134/s0965542506040075.  Google Scholar

[33]

V. T. Volkov and N. N. Nefëdov, Periodic solutions with boundary layers of a singularly perturbed reaction-diffusion model, Computational Mathematics and Mathematical Physics, 34 (1994), 1133-1140.   Google Scholar

[34]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.  Google Scholar

[35]

Y. Xu, Z. J. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized burgers-kdv equation, Nolinear Dynamics, 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

Figure 1.  the picture for the zero approximation $ U_{0}(x,t,\epsilon) $ for the solution $ u(x,t,\epsilon) $ of (50)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Chunyan Ji, Xue Yang, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020122
[2]

Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[3]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[4]

Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure & Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103
[5]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257
[6]

João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[7]

Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[8]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[9]

Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
[10]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89
[11]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020180
[12]

Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[13]

Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
[14]

Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279
[15]

Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[16]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[17]

Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209
[18]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
[19]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[20]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences