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