Polynomial solutions of certain differential equations arising in physics

Abstract

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An algorithm to determine these conditions and to construct the polynomial solutions is given. The effectiveness of this algorithmic approach is illustrated by applying it to several differential equations that arise in mathematical physics.

Several families of new exact solutions for second order partial differential equations with variable coefficients

“…So the coefficients a i are given by the equation ( 8), for 2 i k. Hence the corresponding solution can be written as a homogeneous function of above given degree. This solution is given by equation (9). Once again we will expand the products in the expression given by equation ( 8) using the same values of n i and m i given in the first case, and simplify using Pochhammer symbols to obtain…”

“…where f is a function of the variables x and y and α, β, γ, δ, η, µ, p 1 and p 2 are various parameters. Polynomial solutions of some constant coefficient partial differential equations are discussed in [1][2][3][4][5][6][7][8] and polynomial solutions of some variable coefficient ordinary differential equations are discussed in [9][10][11] and references therein. Exact solutions to the general partial differential equation (1) with variable coefficients are not available in the literature, except for a very few special cases.…”

Higher order self-adjoint operators with polynomial coefficients

“…Classical references that deal with various aspects of polynomial solutions of differential equations are the references [23,7,6,19,16]. More recent papers that deal with the same subject are [14,15,20,25,1,2]. A reference for related topic of orthogonal polynomials is [24].…”