Heat polynomial analogues for equations with higher order time derivatives

Abstract

We generalize the heat polynomials for the heat equation to more general partial differential equations, of higher order with respect to both the time variable and the space variables. Whereas the heat equation requires only one family of polynomials, for an equation of the th order with respect to time we introduce families of polynomials. These families correspond to the initial conditions specified by the Cauchy problem.

Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

“…Proof. Making a particular use of ( 9) with ( 11), ( 6) and applying to Lemma 1, yields our desired result (25) in Theorem 13. The details are omitted.…”

“…Among these classical polynomials are the heat polynomials (also designated as Temperature polynomials) that are polynomial solutions of the heat equation and also are particularly useful in solving the Cauchy problem (see [23][24][25][26]). Special functions, such as the confluent hypergeometric function, integral error functions, and Laguerre polynomials, have a close link with the generalized heat polynomials intro-duced [27][28][29].…”

Analytical method to study a mathematical model of wave processes under two­point time conditions

“…Solutions to problems (10), (7) in the form of (14) and (16) are the quasi-polynomials. Construction of sets of polynomial and quasi-polynomial solutions to partial differential equations and respective boundary problems has been addressed in numerous studies [47][48][49].…”

The differential-symbol method of constructing the quasi-polynomial solutions of two-point problem

“…Also, we note that the action of the differential expression φ (︀ ∂ ∂ν )︀ , symbol of which is the quasi-polynomial φ(x) of the form (15), in formula (16) we understand as follows:…”

“…Many works of scientists [14][15][16] are devoted to constructing the polynomial and quasi-polynomial solutions for the equations and boundary-value problems for them. Let us note that function (4), which is a nontrivial solution to the homogeneous problem (1), (3) is quasi-polynomial, which can be written in the form U(t, x) = Ae αt (e βx + e −βx ) − Ae −αt (e βx + e −βx ),…”