Mixed q-analogues, lacunary polynomials and Truesdellian partial differential equations

global results on some nonlinear partial differential equations for direct and inverse problems

در این رساله به بررسی رفتار جواب های رده ای از معادلات دیفرانسیل با مشتقات جزیی در دامنه های کراندار می پردازیم . این معادلات به فرم نیم-خطی و غیر خطی برای مسایل مستقیم و معکوس مورد مطالعه قرار می گیرند . به ویژه، تاثیر شرایط مختلف فیزیکی را در مساله، نظیر وجود موانع و منابع، پراکندگی و چسبندگی در معادلات موج و گرما بررسی می کنیم و به دنبال شرایطی می گردیم که متضمن وجود سراسری یا عدم وجود سراسر...

Factoring multivariate polynomials via partial differential equations

A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp’s and Niederreiter’s algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to th...

q-analogues of Ehrhart polynomials

One considers weighted sums over points of lattice polytopes, where the weight of a point v is the monomial q for some linear form λ. One proposes a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers.

Fractional differential equations with Legendre polynomials

Orthogonal Polynomials and Partial Differential Equations on the Unit Ball

Orthogonal polynomials of degree n with respect to the weight function Wμ(x) = (1 − ‖x‖2)μ on the unit ball in R are known to satisfy the partial differential equation [ ∆− 〈x,∇〉 − (2μ+ d)〈x,∇〉 ] P = −n(n+ 2μ+ d)P for μ > −1. The singular case of μ = −1,−2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2,−3, . . . is shown to have comple...