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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 equations (ODEs).

Relatit page: Numerical methods for pairtial differential equations

Background

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Syne odes appearit i science, many mathematicians have studiit hou tae solve thaim.[1][2][3][4] 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,[5] 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:

Validatit numerics for ODEs

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Relatit page: 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.[17][18] The popular methods are basit on the shootin method or spectral methods.[19][20] The day, European resairch teams[21][22][23][24][25][26][27][28][29] an Japanese experts[30][31] ar wirkin on this topic.

ODEs an relatit topics studiet in this context

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Relatit saftware

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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.
  • 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.

References

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  1. ↑ Arnolʹd, V. I., Ordinary differential equations. Springer.
  2. ↑ Wolfgang Walter, Ordinary differential equations. Springer.
  3. ↑ Logemann, H., & Ryan, E. P. (2014). Ordinary differential equations: Analysis, qualitative theory and control. Springer.
  4. ↑ Chicone, C. (2006). Ordinary differential equations with applications. Springer Science & Business Media.
  5. ↑ Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied Numerical Mathematics, 20(3), 247-260.
  6. ↑ Monroe, J. L. (2002). Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65(6), 066116.
  7. ↑ Peskin, C. S., & Schlick, T. (1989). Molecular dynamics by the Backward‐Euler method. Communications on pure and applied mathematics, 42(7), 1001-1031.
  8. ↑ Emma Gau (2020). Euler–Maruyama Method (https://www.mathworks.com/matlabcentral/fileexchange/69430-euler-maruyama-method), MATLAB Central File Exchange. Retrieved May 24, 2020.
  9. ↑ Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.
  10. ↑ 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.
  11. ↑ Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer Science & Business Media.
  12. ↑ Symplectic integrators: An introduction, American Journal of Physics 73, 938 (2005); https://doi.org/10.1119/1.2034523 Denis Donnelly.
  13. ↑ Y. B. Suris, Hamiltonian Runge-Kutta type methods and their variational formulation (1990) Matematicheskoe modelirovanie, 2(4), 78-87.
  14. ↑ Iserles, A., & Quispel, G. R. W. (2016). Why geometric integration?. arXiv preprint arXiv:1602.07755.
  15. ↑ Hirayama, H. (2002). Solution of ordinary differential equations by Taylor series method. JSIAM, 12, 1-8.
  16. ↑ 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.
  17. ↑ 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.
  18. ↑ Gidas, B., Ni, W. M., & Nirenberg, L. (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
  19. ↑ Lloyd N. Trefethen (2000) Spectral Methods in MATLAB. SIAM, Philadelphia, PA.
  20. ↑ D. Gottlieb and S. Orzag (1977) "Numerical Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA.
  21. ↑ a b c 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.
  22. ↑ Rihm, R. (1994). Interval methods for initial value problems in ODEs. Topics in Validated Computations, 173-207.
  23. ↑ 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.
  24. ↑ 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.
  25. ↑ Corliss, G. F. (1989). Survey of interval algorithms for ordinary differential equations. Applied Mathematics and Computation, 31, 112-120.
  26. ↑ 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.
  27. ↑ Eijgenraam, P. (1981). The solution of initial value problems using interval arithmetic: formulation and analysis of an algorithm. MC Tracts.
  28. ↑ 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.
  29. ↑ 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.
  30. ↑ 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.
  31. ↑ 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).
  32. ↑ 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.
  33. ↑ Matsue, K., & Takayasu, A. (2019). Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity. arXiv preprint arXiv:1902.01842.
  34. ↑ 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.
  35. ↑ 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.
  36. ↑ 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.
  37. ↑ 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.
  38. ↑ Galias, Z., & Zgliczyński, P. (1998). Computer assisted proof of chaos in the Lorenz equations. Physica D: Nonlinear Phenomena, 115(3-4), 165-188.
  39. ↑ Tucker, W. (1999). The Lorenz attractor exists. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197-1202.
  40. ↑ Zgliczynski, P. (1997). Computer assisted proof of chaos in the Rössler equations and in the Hénon map. Nonlinearity, 10(1), 243.
  41. ↑ Driscoll, T. A., Hale, N., & Trefethen, L. N. (2014). Chebfun guide.
  42. ↑ 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.
  43. ↑ Hashemi, B., & Trefethen, L. N. (2017). Chebfun in three dimensions. SIAM Journal on Scientific Computing, 39(5), C341-C363.
  44. ↑ 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.
  45. ↑ 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.
  46. ↑ Berz, M., Makino, K., Shamseddine, K., Hoffstätter, G. H., & Wan, W. (1996). 32. COSY INFINITY and Its Applications in Nonlinear Dynamics.
  47. ↑ S.M. Rump: INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages 77-104. Kluwer Academic Publishers, Dordrecht, 1999.
  48. ↑ Overview of kv – a C++ library for verified numerical computation, Masahide Kashiwagi, SCAN 2018.
  49. ↑ Wouwer, A. V., Saucez, P., & Vilas, C. (2014). Simulation of Ode/Pde Models with MATLAB®, OCTAVE and Scilab: Scientific and Engineering Applications. Springer.
  50. ↑ Houcque, D. (2008). Applications of MATLAB: Ordinary differential equations (ODE). Robert R. McCormick School of Engineering and Applied Science-Northwestern University, Evanston.
  51. ↑ Shampine, L. F., & Reichelt, M. W. (1997). The matlab ode suite. SIAM Journal on Scientific Computing, 18(1), 1-22.
  52. ↑ Ashino, R., Nagase, M., & Vaillancourt, R. (2000). Behind and beyond the MATLAB ODE suite. Computers & Mathematics with Applications, 40(4-5), 491-512.
  53. ↑ Gladwell, I. (1979). Initial value routines in the NAG library. ACM Transactions on Mathematical Software (TOMS), 5(4), 386-400.
  54. ↑ 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.
  55. ↑ Berzins, M., Brankin, R. W., & Gladwell, I. (1988). The stiff integrators in the NAG library. ACM SIGNUM Newsletter, 23(2), 16-23.
  56. ↑ Baumann, G. (2013). Symmetry analysis of differential equations with Mathematica®. Springer Science & Business Media.
  57. ↑ Abell, M. L., & Braselton, J. P. (2016). Differential equations with Mathematica. Academic Press.
  58. ↑ Gray, A., Mezzino, M., & Pinsky, M. A. (1997). Introduction to ordinary differential equations with Mathematica: an integrated multimedia approach. Springer.
  59. ↑ Ross, C. C. (2013). Differential equations: an introduction with Mathematica®. Springer Science & Business Media.

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