Abstract
A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented.
Highlights
In Pettigrew and Meredith [11], an alternative to the classical solution of the inverse heat conduction problem was derived
The choice of a Laguerre expansion was a logical one for the heat equation and gave rise to an interesting new set of special functions Θn±(x)
It indicates a fresh approach to the inverse Cauchy problem and suggests a host of new expansions of the operators present in the classical solution of this important inverse problem
Summary
A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented. 2000 Mathematics Subject Classification: 05A40, 26C05, 35R25
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