January  2014, 19(1): 281-298. doi: 10.3934/dcdsb.2014.19.281

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

Received  August 2012 Revised  October 2013 Published  December 2013

In this paper, inspired by the work by A. Iserles and G. Söderlind [Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems.
Citation: Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281
References:
[1]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, Ltd 2008. doi: 10.1002/9780470753767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[8]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001.  Google Scholar

[14]

R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004.  Google Scholar

[15]

J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883. doi: 10.1007/BF01954907.  Google Scholar

[16]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[8]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001.  Google Scholar

[14]

R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004.  Google Scholar

[15]

J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883. doi: 10.1007/BF01954907.  Google Scholar

[16]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar

[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 & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021236

[3]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & 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 & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021213

[5]

Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - S, 2019, 12 (6) : 1535-1545. doi: 10.3934/dcdss.2019106

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]