ABSTRACT
Similarly to the correspondence between radical ideals of a polynomial ring and varieties in algebraic geometry, a correspondence between radical differential ideals and their analytic solution sets has been established in differential algebra. This tutorial discusses aspects of this correspondence involving symbolic computation. In particular, an introduction to the Thomas decomposition method is given. It splits a system of polynomially nonlinear partial differential equations into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. The power series solutions of each simple system can be determined in a straightforward way. Conversely, certain sets of analytic functions admit an implicit description in terms of partial differential equations and inequations. Strategies for solving related differential elimination problems and applications to symbolic solving of differential equations are presented. A Maple implementation of the Thomas decomposition method is freely available.
- P. Aubry, D. Lazard, and M. Moreno Maza. On the theories of triangular sets. J. Symb. Comput., 28(1--2):105--124 (1999).Google ScholarCross Ref
- T. Bachler.Counting Solutions of Algebraic Systems via Triangular Decomposition. Ph.D. thesis, RWTH Aachen University, Germany (2014). Available online at http://publications.rwth-aachen.de/record/444946?ln=en.Google Scholar
- T. Bachler and M. Lange-Hegermann. Algebraic Thomas and Differential Thomas: Thomas decomposition of algebraic and differential systems. Available online at http://wwwb.math.rwth-aachen.de/thomasdecomposition.Google Scholar
- T. Bachler, V. P. Gerdt, M. Lange-Hegermann, and D. Robertz.Thomas Decomposition of Algebraic and Differential Systems.In: V. P. Gerdt, W. Koepf, E. W. Mayr, and E. H. Vorozhtsov, editors, Computer Algebra in Scientific Computing, 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia, Vol. 6244 ofLecture Notes in Computer Science, pp. 31--54. Springer, 2010. Google ScholarDigital Library
- T. Bachler, V. P. Gerdt, M. Lange-Hegermann, and D. Robertz.Algorithmic Thomas Decomposition of Algebraic and Differential Systems. J. Symb. Comput., 47(10):1233--1266 (2012). Google ScholarDigital Library
- F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Computing representations for radicals of finitely generated differential ideals. Appl. Algebra Engrg. Comm. Comput., 20(1):73--121 (2009).Google ScholarCross Ref
- X. S. Gao. Implicitization of differential rational parametric equations. J. Symb. Comput., 36(5):811--824 (2003). Google ScholarDigital Library
- V. P. Gerdt. On decomposition of algebraic PDE systems into simple subsystems. Acta Appl. Math., 101(1--3):39--51 (2008).Google ScholarCross Ref
- V. P. Gerdt and D. Robertz. Lagrangian Constraints and Differential Thomas Decomposition. Adv. in Appl. Math., 72:113--138 (2016). Google ScholarDigital Library
- E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems. II. Differential systems.In: F. Winkler and U. Langer, editors, Symbolic and numerical scientific computation (Hagenberg, 2001), pp. 1--39 resp. 40--87,Vol. 2630 ofLecture Notes in Computer Science, Springer, Berlin, 2003. Google ScholarDigital Library
- E. R. Kolchin. Differential algebra and algebraic groups. Vol. 54 of Pure and Applied Mathematics. Academic Press, New York-London, 1973.Google Scholar
- M. Lange-Hegermann.Counting Solutions of Differential Equations.Ph.D. thesis, RWTH Aachen University, Germany (2014). Available online at http://publications.rwth-aachen.de/record/229056?ln=en.Google Scholar
- F. Lemaire, M. Moreno Maza, and Y. Xie. The RegularChains library in MAPLE. SIGSAM Bull., 39:96--97, September 2005. Google ScholarDigital Library
- P. J. Olver and P. Rosenau.The construction of special solutions to partial differential equations. Phys. Lett. A, 114(3):107--112 (1986).Google ScholarCross Ref
- W. Plesken. Counting solutions of polynomial systems via iterated fibrations. Arch. Math. (Basel), 92(1):44--56 (2009).Google ScholarCross Ref
- W. Plesken and D. Robertz.Elimination for coefficients of special characteristic polynomials. Experiment. Math., 17(4):499--510 (2008).Google ScholarCross Ref
- W. Plesken and D. Robertz.Linear differential elimination for analytic functions. Math. Comput. Sci., 4(2--3):231--242 (2010).Google Scholar
- T. M. Rassias and J. Simsa. Finite sums decompositions in mathematical analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Chichester, 1995.Google Scholar
- J. F. Ritt. Differential Algebra. Vol. XXXIII ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, New York, N. Y., 1950.Google Scholar
- D. Robertz. Formal Algorithmic Elimination for PDEs. Vol. 2121 ofLecture Notes in Mathematics, Springer, Cham, 2014.Google Scholar
- S. L. Rueda and J. R. Sendra.Linear complete differential resultants and the implicitization of linear DPPEs. J. Symb. Comput., 45(3):324--341 (2010). Google ScholarDigital Library
- A. Seidenberg. An elimination theory for differential algebra. Univ. California Publ. Math. (N.S.) 3, 31--65 (1956).Google Scholar
- J. M. Thomas. Differential Systems. Vol. XXI ofAmerican Mathematical Society Colloquium Publications.American Mathematical Society, New York, N. Y., 1937.Google Scholar
- D. Wang. Decomposing polynomial systems into simple systems. J. Symb. Comput., 25(3):295--314 (1998). Google ScholarDigital Library
- D. Wang. Elimination methods. Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 2001.Google ScholarCross Ref
- T. Wolf, A. Brand, and M. Mohammadzadeh. Computer algebra algorithms androutines for the computation of conservation laws and fixing of gauge indifferential expressions. J. Symb. Comput., 27(2):221--238 (1999). Google ScholarDigital Library
Index Terms
- Formal Algorithmic Elimination for PDEs
Recommendations
A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics
This paper deals with the numerical solutions of a general class of one-dimensional nonlinear partial differential equations (PDEs) arising in different fields of science. The nonlinear equations contain, as special cases, several PDEs such as Burgers ...
Moving mesh discontinuous Galerkin methods for PDEs with traveling waves
Moving mesh with discontinuous Galerkin method.Nonlinear 1D problems with traveling waves.Resolving sharp moving fronts and determination of correct wave speed.Uncoupling of the discretization and mesh equations. In this paper, a moving mesh ...
The modified (G′/G )- expansion method and its applications to construct exact solutions for nonlinear PDEs
In the present article, we construct the traveling wave solutions involving parameters of some nonlinear PDEs; namely the nonlinear Klein - Gordon equations, the nonlinear reaction- diffusion equation, the nonlinear modified Burgers equation and the ...
Comments