Abstract
Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically.
Similar content being viewed by others
References
Wang S J, Zhang H. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations. Sci China Ser G-Phys Mech Astron, 2006, 49(6): 716–728
Wang S J, Zhang H. Algebraic algebraic dynamics algorithm: Numerical comparison with runge-kutta algorithm and symplectic geometric algorithm. Sci China Ser G-Phys Mech Astron, 2007, 50(1): 1–18
Wang S J, Zhang H. Symplectic algebraic dynamics algorithm. Sci China Ser G-Phys Mech Astron, 2007, 50(2): 133–143
Wang S J, Li F L, Weiguny A. Algebraic dynamics and time-dependent dynamical symmetry of nonautonomous systems. Phys Lett A, 1993, 180: 189
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China
Rights and permissions
About this article
Cite this article
Wang, S., Zhang, H. Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems. Sci. China Ser. G-Phys. Mech. Astron. 51, 577–590 (2008). https://doi.org/10.1007/s11433-008-0055-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11433-008-0055-0