See soorce fer Numerical methods for ordinary differential equations

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'''Numerical methods for ordinary differential equations''' are computational schemes tae obtain approximate solutions o [[ordinary differential equation]]s (ODEs).
{{see also|Numerical methods for pairtial differential equations}}
==Background==
Syne odes appearit i [[science]], many [[mathematician]]s have studiit hou tae solve thaim.<ref>Arnolʹd, V. I., Ordinary differential equations. Springer.</ref><ref>Wolfgang Walter, Ordinary differential equations. Springer.</ref><ref>Logemann, H., & Ryan, E. P. (2014). Ordinary differential equations: Analysis, qualitative theory and control. Springer.</ref><ref>Chicone, C. (2006). Ordinary differential equations with applications. Springer Science & Business Media.</ref> However, only few o thaim can be mathematically solvit. This is why numerical methods are needit. Ane o the most famous methods are the Runge-Kutta methods,<ref>Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathematics, 20(3), 247-260.</ref> but it disnae wirk for some ODEs (especially nonlinear ODEs). This is hou new ode solvers are developit. The followin list includes frequently uised methods:
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* Bulirsch-Stoer algorithm<ref>Monroe, J. L. (2002). Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65(6), 066116.</ref>
* Euler's method (namit after [[Leonhard Euler]])
** Backward Euler method<ref>Peskin, C. S., & Schlick, T. (1989). Molecular dynamics by the Backward‐Euler method. Communications on pure and applied mathematics, 42(7), 1001-1031.</ref>
** Semi-implicit Euler method
** Euler-Maruyama method<ref>Emma Gau (2020). Euler–Maruyama Method (https://www.mathworks.com/matlabcentral/fileexchange/69430-euler-maruyama-method), MATLAB Central File Exchange. Retrieved May 24, 2020.</ref>
* Exponential integrator<ref>Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.</ref><ref>Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integrators. SIAM Journal on Scientific Computing, 33(2), 488-511.</ref>
* Symplectic integrator<ref>Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric [[numerical integration]]: structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media.</ref><ref>Symplectic integrators: An introduction, American Journal of Physics 73, 938 (2005); https://doi.org/10.1119/1.2034523
Denis Donnelly.</ref><ref>Y. B. Suris, Hamiltonian Runge-Kutta type methods and their variational formulation (1990) Matematicheskoe modelirovanie, 2(4), 78-87.</ref><ref>Iserles, A., & Quispel, G. R. W. (2016). Why geometric integration?. arXiv preprint arXiv:1602.07755.</ref>
* Taylor series method<ref>Hirayama, H. (2002). Solution of ordinary differential equations by Taylor series method. JSIAM, 12, 1-8.</ref><ref>Hirayama, H. (2015). Performance of a Higher-Order Numerical Method for Solving Ordinary Differential Equations by Taylor Series. In Integral Methods in Science and Engineering (pp. 321-328). Birkhäuser, Cham.</ref>
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==Validatit numerics for ODEs==
{{see also|Validatit numerics}}
Nae anely approximate solvers, but the study tae "verify the existence o solution bi computers" is also active. This study is needit acause numerically obtaint solutions cud be phantom solutions (fake solutions). This kynd o incident is awreidy reportit.<ref>Breuer, B., Plum, M., & McKenna, P. J. (2001). "Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods." In Topics in Numerical Analysis (pp. 61–77). Springer, Vienna.</ref><ref>Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.</ref> The popular methods are basit on the shootin method or spectral methods.<ref>Lloyd N. Trefethen (2000) Spectral Methods in [[MATLAB]]. [[Society for Industrial and Applied Mathematics|SIAM]], Philadelphia, PA.</ref><ref>D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", [[Society for Industrial and Applied Mathematics|SIAM]], Philadelphia, PA.</ref> The day, European resairch teams<ref name="lohner">Lohner，R．J.，Enclosing the Solution of Ordinary lnitial and Boundary Value Problems, Computer arithmetic：Scientific Computation and Programming Languages，Kaucher，E.，Kulisch，U., Ullrich，Ch．(eds.)， B．G．Teubner，Stuttgart (1987), 255−286.</ref><ref>Rihm, R. (1994). Interval methods for initial value problems in ODEs. Topics in Validated Computations, 173-207.</ref><ref>Hungria, A., Lessard, J. P., & Mireles-James, J. D. (2014). Radii polynomial approach for analytic solutions of differential equations: Theory, examples, and comparisons. Math. Comp.</ref><ref>Nedialkov, N. S., Jackson, K. R., & Pryce, J. D. (2001). An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE. Reliable Computing, 7(6), 449-465.</ref><ref>Corliss, G. F. (1989). Survey of interval algorithms for ordinary differential equations. Applied Mathematics and Computation, 31, 112-120.</ref><ref>Nedialkov, N. S. (2000). Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation (Ph.D. thesis). University of Toronto.</ref><ref>Eijgenraam, P. (1981). The solution of initial value problems using interval arithmetic: formulation and analysis of an algorithm. MC Tracts.</ref><ref>Nedialkov, N. S., & Jackson, K. R. (1999). An interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Reliable Computing, 5(3), 289-310.</ref><ref>Nedialkov, N. S., Jackson, K. R., & Corliss, G. F. (1999). Validated solutions of initial value problems for ordinary differential equations. Applied Mathematics and Computation, 105(1), 21-68.</ref> an Japanese experts<ref>Berz, M., & Makino, K. (1998). Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable computing, 4(4), 361-369.</ref><ref>Kashiwagi, M. (1995). Numerical Validation for Ordinary Differential Equations using Power Series Arithmetic. In Numerical Analysis Of Ordinary Differential Equations And Its Applications (pp. 213-218).</ref> ar wirkin on this topic.
===ODEs an relatit topics studiet in this context===
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* Blow-up solutions<ref>Takayasu, A., Matsue, K., Sasaki, T., Tanaka, K., Mizuguchi, M., & Oishi, S. I. (2017). Numerical validation of blow-up solutions of ordinary differential equations. Journal of Computational and Applied Mathematics, 314, 10-29.</ref><ref>Matsue, K., & Takayasu, A. (2019). Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity. arXiv preprint arXiv:1902.01842.</ref>
* Lorentz equation<ref>Hassard, B., Zhang, J., Hastings, S. P., & Troy, W. C. (1994). A computer proof that the Lorenz equations have “chaotic” solutions. Applied Mathematics Letters, 7(1), 79-83.</ref><ref>Mischaikow, K., & Mrozek, M. (1995). Chaos in the Lorenz equations: a computer-assisted proof. [[:en:Bulletin of the American Mathematical Society]], 32(1), 66-72.</ref><ref>Mischaikow, K., & Mrozek, M. (1998). Chaos in the Lorenz equations: A computer assisted proof. Part II: Details. [[:en:Mathematics of Computation]], 67(223), 1023-1046.</ref><ref>Mischaikow, K., Mrozek, M., & Szymczak, A. (2001). Chaos in the lorenz equations: A computer assisted proof part iii: Classical parameter values. Journal of Differential Equations, 169(1), 17-56.</ref><ref>Galias, Z., & Zgliczyński, P. (1998). Computer assisted proof of chaos in the Lorenz equations. Physica D: Nonlinear Phenomena, 115(3-4), 165-188.</ref><ref>Tucker, W. (1999). The Lorenz attractor exists. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197-1202.</ref>
* Rossler equation<ref>Zgliczynski, P. (1997). Computer assisted proof of chaos in the Rössler equations and in the Hénon map. Nonlinearity, 10(1), 243.</ref>
===Relatit saftware===
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*Chebfun<ref>Driscoll, T. A., Hale, N., & Trefethen, L. N. (2014). Chebfun guide.</ref><ref>
Platte, R. B., & Trefethen, L. N. (2010). Chebfun: a new kind of numerical computing. In Progress in industrial mathematics at ECMI 2008 (pp. 69-87). Springer, Berlin, Heidelberg.</ref><ref>Hashemi, B., & Trefethen, L. N. (2017). Chebfun in three dimensions. SIAM Journal on Scientific Computing, 39(5), C341-C363.</ref><ref>Wright, G. B., Javed, M., Montanelli, H., & Trefethen, L. N. (2015). Extension of Chebfun to periodic functions. SIAM Journal on Scientific Computing, 37(5), C554-C573.</ref>
*[https://www.bt.pa.msu.edu/index_cosy.htm COSY INFINITY] {{Webarchive|url=https://web.archive.org/web/20200924140241/https://bt.pa.msu.edu/index_cosy.htm |date=2020-09-24 }}<ref>Makino, K., & Berz, M. (2006). Cosy infinity version 9. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 558(1), 346-350.</ref><ref>Berz, M., Makino, K., Shamseddine, K., Hoffstätter, G. H., & Wan, W. (1996). 32. COSY INFINITY and Its Applications in Nonlinear Dynamics.</ref>
* [[INTLAB]]<ref name="lohner"/><ref>S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.</ref> and kv<ref name="lohner"/><ref>Overview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.</ref> - [[interval arithmetic]] libraries includin ODE solvers
* [[MATLAB]]<ref>Wouwer, A. V., Saucez, P., & Vilas, C. (2014). Simulation of Ode/Pde Models with MATLAB®, [[GNU Octave|OCTAVE]] and [[Scilab]]: Scientific and Engineering Applications. Springer.</ref><ref>Houcque, D. (2008). Applications of MATLAB: Ordinary differential equations (ODE). Robert R. McCormick School of Engineering and Applied Science-Northwestern University, Evanston.</ref><ref>Shampine, L. F., & Reichelt, M. W. (1997). The matlab ode suite. SIAM Journal on Scientific Computing, 18(1), 1-22.</ref><ref>Ashino, R., Nagase, M., & Vaillancourt, R. (2000). Behind and beyond the MATLAB ODE suite. Computers & Mathematics with Applications, 40(4-5), 491-512.</ref>
* NAG library<ref>Gladwell, I. (1979). Initial value routines in the NAG library. ACM Transactions on Mathematical Software (TOMS), 5(4), 386-400.</ref><ref>Gladwell, I. (1979). The development of the boundary-value codes in the ordinary differential equations chapter of the NAG library. In Codes for Boundary-Value Problems in Ordinary Differential Equations (pp. 122-143). Springer, Berlin, Heidelberg.</ref><ref>Berzins, M., Brankin, R. W., & Gladwell, I. (1988). The stiff integrators in the NAG library. ACM SIGNUM Newsletter, 23(2), 16-23.</ref>
* [[Wolfram Mathematica]]<ref>Baumann, G. (2013). Symmetry analysis of differential equations with Mathematica®. Springer Science & Business Media.</ref><ref>Abell, M. L., & Braselton, J. P. (2016). Differential equations with Mathematica. Academic Press.</ref><ref>Gray, A., Mezzino, M., & Pinsky, M. A. (1997). Introduction to ordinary differential equations with Mathematica: an integrated multimedia approach. Springer.</ref><ref>Ross, C. C. (2013). Differential equations: an introduction with Mathematica®. Springer Science & Business Media.</ref>

==Forder readin==
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* Mitsui, T., & Shinohara, Y. (1995). Numerical analysis of ordinary differential equations and its applications. World Scientific.
* Iserles, A. (2009). A first course in the numerical analysis of differential equations. [[Cambridge University Press]].
* Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, [[Berlin]], [[New York]]: Springer-Verlag.
* Wanner, G. & Hairer, E. (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.). Springer Berlin Heidelberg.
* Butcher, John C. (2008), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons.
* John D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
* [[Peter Deuflhard|Deuflhard, P.]], & Bornemann, F. (2012). Scientific computing with ordinary differential equations. Springer Science & Business Media.
* Shampine, L. F. (2018). Numerical solution of ordinary differential equations. Routledge.
* Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press.
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==References==
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==Fremmit airtins==
* Joseph W. Rudmin, ''[http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf Application of the Parker–Sochacki Method to Celestial Mechanics] {{Webarchive|url=http://arquivo.pt/wayback/20160516155343/http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf |date=2016-05-16 }}'', 1998.
* Dominique Tournès, ''[https://web.archive.org/web/20130413090625/http://www.reunion.iufm.fr/dep/mathematiques/calculsavant/Equipe/tournes.html L'intégration approchée des équations différentielles ordinaires (1671-1914)]'', thèse de doctorat de l'université Paris 7 - Denis Diderot, juin 1996. Réimp. Villeneuve d'Ascq : Presses universitaires du Septentrion, 1997, 468 p.
*[http://www.ti3.tu-harburg.de/rump/intlab/ INTLAB] {{Webarchive|url=https://web.archive.org/web/20200130222824/http://www.ti3.tu-harburg.de/rump/intlab/ |date=2020-01-30 }}
*[http://verifiedby.me/kv/demo/ode.cgi Verified ODE (IVP) Solver]
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