The Exact Solutions for Several Partial Differential-Difference Equations with Constant Coefficients

Hongyan Xu (), Ling Xu and Hari Mohan Srivastava
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Hongyan Xu: College of Arts and Sciences, Suqian University, Suqian 223800, China
Ling Xu: School of Mathematics and Computer, Jiangxi Science and Technology Normal University, Nanchang 330038, China
Hari Mohan Srivastava: Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada

Mathematics, 2022, vol. 10, issue 19, 1-15

Abstract: This article is concerned with the description of the entire solutions of several Fermat type partial differential-difference equations (PDDEs) μ f ( z ) + λ f z 1 ( z ) 2 + [ α f ( z + c ) − β f ( z ) ] 2 = 1 , and μ f ( z ) + λ 1 f z 1 ( z ) + λ 2 f z 2 ( z ) 2 + [ α f ( z + c ) − β f ( z ) ] 2 = 1 , where f z 1 ( z ) = ∂ f ∂ z 1 and f z 2 ( z ) = ∂ f ∂ z 2 , c = ( c 1 , c 2 ) ∈ C 2 , α , β , μ , λ , λ 1 , λ 2 , c 1 , c 2 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.

Keywords: Nevanlinna theory; entire solution; partial differential-difference equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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