The exact solutions for several partial differential-difference equations with constant coefficients
Date
2022
Authors
Xu, Hongyan
Xu, Ling
Srivastava, H.M.
Journal Title
Journal ISSN
Volume Title
Publisher
Mathematics
Abstract
This article is concerned with the description of the entire solutions of several Fermat
type partial differential-difference equations (PDDEs) [μf(z) + λfz1(z)]^2 + [αf(z + c) - βf(z)]^2 = 1 and [μf(z) + λ1fz1(z) + λ2fz2(z)]^2 + [αf(z + c) - βf(z)]^2 = 1, where fz1(z) = ∂f/∂z1 and fz2(z) = ∂f/∂z2, c = (c1,c2) ∈ ℂ^2, α,β,μ,λ,λ1,λ2,c1,c2 are constants in ℂ. Our theorems in this paper give some
descriptions of the forms of transcendental entire solutions for the above equations, which are some
extensions and improvement of the previous theorems given by Xu, Cao, Liu, and Yang. In particular,
we exhibit a series of examples to explain that the existence conditions and the forms of transcendental
entire solutions with a finite order of such equations are precise.
Description
Keywords
Nevanlinna theory, entire solution, partial differential-difference equation
Citation
Xu, H., Xu, L., & Srivastava, H. M. (2022). “The exact solutions for several partial differential-difference equations with constant coefficients.” Mathematics, 10(19), 3596. https://doi.org/10.3390/math10193596