A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

[1]  Franz Winkler,et al.  Rational general solutions of first order non-autonomous parametrizable ODEs , 2010, J. Symb. Comput..

[2]  A. Cañada,et al.  Handbook of differential equations , 2004 .

[3]  Josef Schicho,et al.  Rational Parametrization of Surfaces , 1998, J. Symb. Comput..

[4]  L. X. C. Ngô.  Birational Transformations on Algebraic Ordinary Differential Equations , 2012 .

[5]  Xiao-Shan Gao,et al.  A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs , 2006, J. Symb. Comput..

[6]  S. B Engelsman.  Lagrange's early contributions to the theory of first-order partial differential equations , 1980 .

[7]  M. V. Safonov,et al.  Lectures on Partial Differential Equations , 2014 .

[8]  Franz Winkler,et al.  Rational general solutions of parametrizable AODEs , 2011 .

[9]  Franz Winkler,et al.  Polynomial Algorithms in Computer Algebra , 1996, Texts and Monographs in Symbolic Computation.

[10]  Franz Winkler,et al.  Rational General Solutions of Trivariate Rational Differential Systems , 2012, Math. Comput. Sci..

[11]  I. Petrovsky,et al.  Lectures On Partial Differential Equations , 1962 .

[12]  Xiao-Shan Gao,et al.  Rational general solutions of algebraic ordinary differential equations , 2004, ISSAC '04.

[13]  Alexandre Eremenko.  RATIONAL SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS , 1998 .

[14]  Franz Winkler,et al.  Rational general solutions of higher order algebraic odes , 2013, J. Syst. Sci. Complex..

[15]  Howard Whitley Eves,et al.  Elementary Matrix Theory , 1980 .

[16]  Georg Grasegger.  Radical solutions of first order autonomous algebraic ordinary differential equations , 2014, ISSAC.

[17]  Franz Winkler,et al.  Classification of algebraic ODEs with respect to rational solvability , 2012 .

[18]  Georg Grasegger.  A procedure for solving autonomous AODEs , 2013 .

[19]  J. Rafael Sendra,et al.  First steps towards radical parametrization of algebraic surfaces , 2012, Comput. Aided Geom. Des..

[20]  Franz Winkler,et al.  Rational general solutions of planar rational systems of autonomous ODEs☆ , 2011, J. Symb. Comput..

[21]  Wolfgang Arendt,et al.  Partielle Differenzialgleichungen: Eine Einführung in analytische und numerische Methoden , 2018 .

[22]  J. Rafael Sendra,et al.  Birational transformations preserving rational solutions of algebraic ordinary differential equations , 2015, J. Comput. Appl. Math..

[23]  J. Rafael Sendra,et al.  On Symbolic Solutions of Algebraic Partial Differential Equations , 2014, CASC.

[24]  Evelyne Hubert,et al.  The general solution of an ordinary differential equation , 1996, ISSAC '96.

[25]  Franz Winkler,et al.  Symbolic Solutions of First-Order Algebraic ODEs , 2015, Computer Algebra and Polynomials.

[26]  E. Kamke.  Differentialgleichungen : Lösungsmethoden und Lösungen , 1977 .

[27]  Friedemann Mattern.  Modellbildung und Simulation , 1993, Perspektiven der Informatik.