Abstract
We give the expansion of the powers of the Lie operator Δ = Σλ i ∂ i in any dimension, whereλ is either a smooth function or a formal power series over an infinite set of commutatives indeterminates. We describe an algorithm for computer treatment and we give, as an example, a table for the orders 1 to 6.
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References
Ginocchio, M. and Irac-Astaud, M., A recursive linearization process for evolution equations...,Rep. Math. Phys. 21, 245–265 (1985).
Steeb, W. H. and Euler, N.,Nonlinear Evolution Equations and Painlevé Test, World Scientific, Singapore, 1988.
Comtet, L., Une formule explicite pour les puissances successives de l'opérateur de dérivation de Lie,C. R. Acad. Sci. Paris A 276, 165–168 (1973).
Leroux, P. and Viennot, G., Combinatorial resolution of systems of differential equations I: ordinary differential equations, in C. Labelle and P. Leroux (eds),Actes du colloque de combinatoire énumérative, Montreal 1985 Lecture Notes in Math. 1234, Springer, New York.
Bergeron, F. and Reutenauer, C., Une interprétation combinatoire des puissances d'un opérateur differentiel linéaire,Ann. Sci. Math. Quebec. 11, 269–278 (1987).
Grossman, R. and Larson, R. G., (a) Hopf-algebraic structures of families of trees,J. Algebra 126, 184–210 (1989); (b) Solving nonlinear equations from higher order derivations in linear stages,Adv. in Math. 82, 180–202 (1990).
Ginocchio, M., On the bi-algebras of functional graphs, linear order extensions and differential algebras, in preparation.
Butcher, J. C.,The Numerical Analysis of Ordinary Differential Equations, Wiley, New York, 1986.
Harary, F. and Prins, G., The number of homeomorphically irreducible trees and other species,Acta Math. 101, 141–162 (1959).
Chiricota, Y.,Structures combinatoires et calcul symbolique, Publ. 12, LACIM, Montreal, 1992.
Stanley, R. P.,Enumerative Combinatorics, Vol. 1, Wadsworth, Monterey, 1986.
Ginocchio, M., Functionals and symmetric functions, application to nonlinear evolution equations, in preparation.