American Institute of Mathematical Sciences

Advanced
November  2003, 3(4): 495-504. doi: 10.3934/dcdsb.2003.3.495

Using numerical experiments to discover theorems in differential equations

Joseph D. Fehribach 1,

1. 

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2247, United States

Received  December 2002 Revised  March 2003 Published  August 2003

This work explores the use of numerical experiments in two specific cases: (1) the discovery of two families of exact solutions to the elastic string equations, and approximately periodic solutions that appear to exist near pseudo-solutions formed from these families; (2) the study of the diffusion-reaction-conduction process in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. This latter work establishes the well-posedness of the electrolyte wedge problem and provides asymptotic expansions for the current density and total current produced by such a wedge. The theme of this paper is the use of computing to discover a result that is difficult or impossible to find without a computer, but which once observed, can then be proven mathematically.
Keywords: Numerical experiments, maximum principle, electrolyte wedge problem, matched asymptotics., exact solutions, elastic string, periodic pseudosolutions, energy.
Mathematics Subject Classification: 35L65, 74J30, 35J2.
Citation: Joseph D. Fehribach. Using numerical experiments to discover theorems in differential equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 495-504. doi: 10.3934/dcdsb.2003.3.495
[1]

Björn Gebhard. Periodic solutions for the N-vortex problem via a superposition principle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5443-5460. doi: 10.3934/dcds.2018240
[2]

Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007
[3]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[4]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[5]

Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116
[6]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[7]

Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations & Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024
[8]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[9]

Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079
[10]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[11]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090
[12]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[13]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[14]

Alexei Heintz, Andrey Piatnitski. Osmosis for non-electrolyte solvents in permeable periodic porous media. Networks & Heterogeneous Media, 2016, 11 (3) : 471-499. doi: 10.3934/nhm.2016005
[15]

Elena Beretta, Eric Bonnetier, Elisa Francini, Anna L. Mazzucato. Small volume asymptotics for anisotropic elastic inclusions. Inverse Problems & Imaging, 2012, 6 (1) : 1-23. doi: 10.3934/ipi.2012.6.1
[16]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[17]

Cheng Hou Tsang, Boris A. Malomed, Kwok Wing Chow. Exact solutions for periodic and solitary matter waves in nonlinear lattices. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1299-1325. doi: 10.3934/dcdss.2011.4.1299
[18]

Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697
[19]

G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783
[20]

Gabriella Bretti, Maya Briani, Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 379-394. doi: 10.3934/dcdss.2014.7.379

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences