-
Previous Article
Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations
- DCDS-B Home
- This Issue
-
Next Article
A dynamics approach to a low-order climate model
Dirichlet series for dynamical systems of first-order ordinary differential equations
1. | School of Mathematics & Physics, Qingdao University of Science & Technology, Qingdao 266061, China |
2. | Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, United Kingdom |
References:
[1] |
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, Ltd 2008.
doi: 10.1002/9780470753767. |
[2] |
Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems, Phys. Lett. A, 372 (2008), 6551-6559.
doi: 10.1016/j.physleta.2008.09.014. |
[3] |
A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators, Numer. Math., 82 (1999), 635-646.
doi: 10.1007/s002110050434. |
[4] |
E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12 (2003), 399-450.
doi: 10.1017/S0962492902000144. |
[5] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993. |
[6] |
G. H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge Tracts in Mathematics and Mathematical Physics, 18 Stechert-Hafner, Inc., New York (1964). |
[7] |
M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090.
doi: 10.1137/040611434. |
[8] |
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008. |
[9] |
A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations, BIT, 38 (1998), 315-346.
doi: 10.1007/BF02512370. |
[10] |
A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112.
doi: 10.1006/jcom.1993.1007. |
[11] |
A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods, J. Comp. Appl. Maths., 125 (2000), 69-81.
doi: 10.1016/S0377-0427(00)00459-3. |
[12] |
S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972. |
[13] |
L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001. |
[14] |
R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004. |
[15] |
J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883.
doi: 10.1007/BF01954907. |
[16] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-642-97149-5. |
show all references
References:
[1] |
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, Ltd 2008.
doi: 10.1002/9780470753767. |
[2] |
Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems, Phys. Lett. A, 372 (2008), 6551-6559.
doi: 10.1016/j.physleta.2008.09.014. |
[3] |
A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators, Numer. Math., 82 (1999), 635-646.
doi: 10.1007/s002110050434. |
[4] |
E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12 (2003), 399-450.
doi: 10.1017/S0962492902000144. |
[5] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993. |
[6] |
G. H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge Tracts in Mathematics and Mathematical Physics, 18 Stechert-Hafner, Inc., New York (1964). |
[7] |
M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090.
doi: 10.1137/040611434. |
[8] |
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008. |
[9] |
A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations, BIT, 38 (1998), 315-346.
doi: 10.1007/BF02512370. |
[10] |
A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112.
doi: 10.1006/jcom.1993.1007. |
[11] |
A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods, J. Comp. Appl. Maths., 125 (2000), 69-81.
doi: 10.1016/S0377-0427(00)00459-3. |
[12] |
S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972. |
[13] |
L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001. |
[14] |
R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004. |
[15] |
J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883.
doi: 10.1007/BF01954907. |
[16] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-642-97149-5. |
[1] |
William Guo. Unification of the common methods for solving the first-order linear ordinary differential equations. STEM Education, 2021, 1 (2) : 127-140. doi: 10.3934/steme.2021010 |
[2] |
Yu Guo, Xiao-Bao Shu, Qianbao Yin. Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021236 |
[3] |
Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065 |
[4] |
Adolfo Damiano Cafaro, Simone Fiori. Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3947-3969. doi: 10.3934/dcdsb.2021213 |
[5] |
Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations and Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355 |
[6] |
Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693 |
[7] |
Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029 |
[8] |
Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353 |
[9] |
Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225 |
[10] |
Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games. Networks and Heterogeneous Media, 2020, 15 (4) : 681-710. doi: 10.3934/nhm.2020019 |
[11] |
Gábor Kiss, Bernd Krauskopf. Stability implications of delay distribution for first-order and second-order systems. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 327-345. doi: 10.3934/dcdsb.2010.13.327 |
[12] |
Levon Nurbekyan. One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 963-990. doi: 10.3934/dcdss.2018057 |
[13] |
Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 |
[14] |
W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 |
[15] |
Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040 |
[16] |
Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 |
[17] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
[18] |
Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419 |
[19] |
Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173 |
[20] |
Sylvia Anicic. Existence theorem for a first-order Koiter nonlinear shell model. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1535-1545. doi: 10.3934/dcdss.2019106 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]