126,585 research outputs found
On polynomial solutions of differential equations
A general method of obtaining linear differential equations having polynomial
solutions is proposed. The method is based on an equivalence of the spectral
problem for an element of the universal enveloping algebra of some Lie algebra
in the "projectivized" representation possessing an invariant subspace and the
spectral problem for a certain linear differential operator with variable
coefficients. It is shown in general that polynomial solutions of partial
differential equations occur; in the case of Lie superalgebras there are
polynomial solutions of some matrix differential equations, quantum algebras
give rise to polynomial solutions of finite--difference equations.
Particularly, known classical orthogonal polynomials will appear when
considering acting on . As examples, some
polynomials connected to projectivized representations of ,
, and are briefly discussed.Comment: 12p
Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations
A new algorithm is presented to find exact traveling wave solutions of
differential-difference equations in terms of tanh functions. For systems with
parameters, the algorithm determines the conditions on the parameters so that
the equations might admit polynomial solutions in tanh.
Examples illustrate the key steps of the algorithm. Parallels are drawn
through discussion and example to the tanh-method for partial differential
equations.
The new algorithm is implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute traveling wave
solutions of nonlinear polynomial differential-difference equations. Use of the
package, implementation issues, scope, and limitations of the software are
addressed.Comment: 19 pages submitted to Computer Physics Communications. The software
can be downloaded at http://www.mines.edu/fs_home/wherema
Differential Galois Theory of Linear Difference Equations
We present a Galois theory of difference equations designed to measure the
differential dependencies among solutions of linear difference equations. With
this we are able to reprove Hoelder's Theorem that the Gamma function satisfies
no polynomial differential equation and are able to give general results that
imply, for example, that no differential relationship holds among solutions of
certain classes of q-hypergeometric functions.Comment: 50 page
On a moment generalization of some classical second-order differential equations generating classical orthogonal polynomials
The aim of the work is to construct new polynomial systems, which are
solutions to certain functional equations which generalize the second-order
differential equations satisfied by the so called classical orthogonal
polynomial families of Jacobi, Laguerre, Hermite and Bessel. These functional
equations can be chosen to be of different type: fractional differential
equations, q-difference equations, etc, which converge to their respective
differential equations of the aforesaid classical orthogonal polynomials. In
addition to this, there exists a confluence of both the families of polynomials
constructed and the functional equations who approach to the classical families
of polynomials and second-order differential equations, respectivel
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