14,514 research outputs found

    Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

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    Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi's elliptic functions. For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples illustrate key steps of the algorithms. The new algorithms are implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed. A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at http://www.mines.edu/fs_home/whereman

    Thomas decompositions of parametric nonlinear control systems

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    This paper presents an algorithmic method to study structural properties of nonlinear control systems in dependence of parameters. The result consists of a description of parameter configurations which cause different control-theoretic behaviour of the system (in terms of observability, flatness, etc.). The constructive symbolic method is based on the differential Thomas decomposition into disjoint simple systems, in particular its elimination properties

    Differentiable Genetic Programming

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    We introduce the use of high order automatic differentiation, implemented via the algebra of truncated Taylor polynomials, in genetic programming. Using the Cartesian Genetic Programming encoding we obtain a high-order Taylor representation of the program output that is then used to back-propagate errors during learning. The resulting machine learning framework is called differentiable Cartesian Genetic Programming (dCGP). In the context of symbolic regression, dCGP offers a new approach to the long unsolved problem of constant representation in GP expressions. On several problems of increasing complexity we find that dCGP is able to find the exact form of the symbolic expression as well as the constants values. We also demonstrate the use of dCGP to solve a large class of differential equations and to find prime integrals of dynamical systems, presenting, in both cases, results that confirm the efficacy of our approach

    Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations

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    Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to polynomial systems of ordinary and partial differential equations. The second and third algorithms allow one to explicitly compute polynomial conserved densities and higher-order symmetries of nonlinear evolution and lattice equations. The first algorithm is implemented in the symbolic syntax of both Macsyma and Mathematica. The second and third algorithms are available in Mathematica. The codes can be used for computer-aided integrability testing of nonlinear differential and lattice equations as they occur in various branches of the sciences and engineering. Applied to systems with parameters, the codes can determine the conditions on the parameters so that the systems pass the Painlev\'e test, or admit a sequence of conserved densities or higher-order symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style files elsart.sty and elsart12.st
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