Symbolic Software for the Investigation of Nonlinear Partial Differential Equations

用于研究非线性偏微分方程的符号软件

• 批准号: 9625421
• 负责人: Willy Hereman
• 金额: $ 9.2万
• 依托单位: Colorado School of Mines
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-09-15 至 1999-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625421&HistoricalAwards=false
• 关键词: Symbolic; Software; Investigation; Nonlinear; Partial

This project continues the development and implementation of new powerful symbolic algorithms for testing the integrability and the calculation of conservation laws. The construction of closed- form solutions for systems of nonlinear partial differential equations (PDEs) will also be investigated. In particular, for systems of nonlinear evolution equations, a first symbolic software package will automatically generate polynomial-type conservation laws. Based on scaling properties (symmetries) of the equations, the proposed algorithm provides an elegant way to construct conserved densities and their associated fluxes. Based on homogenization techniques, a second software package will allow the user to compute the analytic form of particular solutions for large classes of PDEs. For integrable equations, the technique leads to soliton solutions. For non-integrable equations, closed-form expressions of homoclinic and heteroclinic orbits can be computed efficiently with the method. The symbolic software to be developed will complement and build on earlier algorithms for the computation of Lie-point and non- classical symmetries, to find soliton solutions via Hirota's direct method, and to test the integrability of differential equations via the Painleve test. The programs will be written in Macsyma and Mathematica syntax. They will be entirely symbolic and produce analytical output. The development of novel mathematical techniques, in addition to the refinement and generalization of existing methods and algorithms, is also a part of this research project. The aim is to provide high-quality symbolic packages to researchers working on soliton theory, dynamical systems, mathematical physics; specifically nonlinear wave phenomena in networks, elastic media, chemical kinetics, bio-sciences, fluid dynamics, plasma physics, and nonlinear optics. ***

该项目继续开发和实施新的强大的符号算法，用于测试可积性和计算守恒律。 还将研究非线性偏微分方程（PDE）系统的封闭形式解的构造。 特别地，对于非线性发展方程组，第一符号软件包将自动生成多项式型守恒律。基于方程的标度性质（对称性），该算法提供了一种优雅的方式来构建守恒密度及其相关通量。 基于均匀化技术，第二个软件包将允许用户计算大类偏微分方程的特定解的解析形式。对于可积方程，该技术导致孤立子解决方案。对于不可积方程，该方法可以有效地计算同宿轨道和异宿轨道的封闭形式。 将要开发的符号软件将补充并建立在用于计算李点和非经典对称性的早期算法的基础上，通过Hirota的直接方法找到孤子解，并通过Painleve测试来测试微分方程的可积性。 程序将以Macsyma和Mathematica语法编写。 它们将完全是象征性的，并产生分析输出。 新的数学技术的发展，除了现有的方法和算法的细化和推广，也是这个研究项目的一部分。 其目的是为研究孤子理论，动力系统，数学物理的研究人员提供高质量的符号包;特别是网络中的非线性波动现象，弹性介质，化学动力学，生物科学，流体动力学，等离子体物理和非线性光学。 ***