本文提出了一种新的构造非线性演化方程行波解的并行算法.我们在Maple 18上实现了该算法.通过设计并行算法并使用负载均衡技术，其中的软件PREM的计算效率明显高于已有的串行软件.且基于因式分解算法和运行时间限制，PREM可以自动推导出一些串行程序算不动的复杂方程的部分精确解.相比于已有的其他程序，PREM可自动推导出更多类型的精确行波解.此外，PREM具有灵活的接口和输出.
In this paper, a new parallel algorithm and its implementation called PREM for solving nonlinear evolution equations are presented. PREM is developed in Maple 18. By using parallel and load balancing techniques, PREM is more efficient than any previous serial programs. Furthermore, for some complicated equations that serial programs failed to solve, PREM may obtain some exact traveling wave solutions by factoring algorithm and time limit. In addition, the interface and output of PREM is flexible and diverse. More types of exact travelling wave solutions could be obtained by using this parallel program.
[1] PARKES E J, DUFFY B R. An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations [J]. Computer Physics Communications, 1996, 98(3): 288-300.
[2] EL S A. Modified extended tanh-function method for solving nonlinear partial differential equations [J]. Chaos Solitons & Fractals, 2007, 31(5): 1256-1264.
[3] ABDOU M A. The extended tanh method and its applications for solving nonlinear physical models [J]. Applied Mathematics & Computation, 2007, 190(1): 988-996.
[4] LIU S K, FU Z T, LIU S D, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations [J]. Physics Letters A, 2001, 289: 69-74.
[5] FU Z T, LIU S K, LIU S D, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations [J]. Physics Letters A, 2001, 290(1/2): 72-76.
[6] YAN Z. The extended Jacobian elliptic function expansion method and its application in the generalized Hirota-Satsuma coupled KdV system [J]. Chaos Solitons & Fractals, 2003, 15(3): 575-583.
[7] HE J H, WU X H. Exp-function method for nonlinear wave equations [J]. Chaos Solitons & Fractals, 2006, 30(3): 700-708.
[8] MA W X, HUANG T, YI Z. A multiple exp-function method for nonlinear differential equations and its appli-cation [J]. Physica Scripta, 2010, 82(6): 5468-5478.
[9] LI Z B, LIU Y P. RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations [J]. Computer Physics Communications, 2002, 148(2): 256-266.
[10] LIU Y P, LI Z B. A Maple package for finding exact solitary wave solutions of coupled nonlinear evolution equations [J]. Computer Physics Communications, 2003, 155(1): 65-76.
[11] LIU Y P, LI Z B. An automated Jacobi elliptic function method for finding periodic wave solutions to nonlinear evolution equation [J]. Chinese Physics Letters, 2002, 19(9): 1228-1230.
[12] BALDWIN D, GOKTAS U, HEREMAN W, et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs [J]. Journal of Symbolic Computation, 2004, 37(6): 669-705.
[13] LI Z B, LIU Y P. RAEEM: A Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations [J]. Computer Physics Communications, 2004, 163(3): 191-201.
[14] ZHANG J X, GU Z M, ZHENG C. Survey of research progress on cloud computing [J]. Application Research of Computers, 2010, 27(2): 429-433.
[15] BERA S, MISRA S, RODRIGUES J J P C. Cloud computing applications for smart grid: A Survey [J]. IEEE Transactions on Parallel & Distributed Systems, 2015, 26(5): 1477-1494.
[16] OWENS J D, LUEBKE D, GOVINDARAJU N, et al. A survey of general-purpose computation on graphics hardware [J]. Computer Graphics Forum, 2007, 26(1): 80-113.
[17] CANT-PAZ E. A survey of parallel genetic algorithms [J]. Calculateurs Paralleles Reseaux Et Systems Repartis, 1999, 10(4): 429-449.
[18] UPADHYAYA S R. Parallel approaches to machine learning——A comprehensive survey [J]. Journal of Parallel & Distributed Computing, 2013, 73(3): 284-292.
[19] AGRAWAL R, SHAFER J C. Parallel mining of association rules [J]. Knowledge & Data Engineering IEEE Transactions on, 1996, 8(6): 962-969.
[20] BERGHEN F V, BERSINI H. CONDOR, a new parallel, constrained extension of Powell's UOBYQA algo-rithm: Experimental results and comparison with the DFO algorithm [J]. Journal of Computational & Applied Mathematics, 2005, 181(1): 157-175.
[21] CHALABINE M, KESSLER C. A Survey of reasoning in parallelization [C]//Proceedings of the Eighth Acis International Conference on Software Engineering. IEEE Computer Society, 2007(3): 629-634.
[22] WING O, HUANG J W. A computation model of parallel solution of linear equations [J]. IEEE Transactions on Computers, 1980, 29(7): 632-638.
[23] LI G J, WAH B W. Computational efficiency of parallel combinatorial or-tree searches [J]. IEEE Transactions on Software Engineering, 1990, 16(1): 13-31.
[24] HOFFMANN K H, ZOU J. Parallel solution of variational inequality problems with nonlinear source terms [J]. Ima Journal of Numerical Analysis, 1996, 16(1): 31-45.
[25] GALIL Z, PAN V. Parallel evaluation of the determinant and of the inverse of a matrix [J]. Information Processing Letters, 1989, 30(1): 41-45.
[26] WANGWQ. A parallel alternating difference implicit scheme for a dispersive equation[J].Mathematica Numerica Sinica, 2005, 27(2): 129-140.
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