Results on the solutions of several second order mixed type partial differential difference equations

Wenju Tang; Keyu Zhang; Hongyan Xu; Wenju Tang; Keyu Zhang; Hongyan Xu

doi:10.3934/math.2022110

AIMS Mathematics

2022, Volume 7, Issue 2: 1907-1924. doi: 10.3934/math.2022110

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Research article

Results on the solutions of several second order mixed type partial differential difference equations

Wenju Tang ¹,
Keyu Zhang ^{2
,
,},
Hongyan Xu ^{3
,
,}

1.
School of Informatics and Engineering, Jingdezhen Ceramic University, Jingdezhen, Jiangxi, 333403, China
2.
School of mathematics, Qilu Normal university, Jinan, Shandong, 250200, China
3.
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao Jiangxi, 334001, China

Received: 08 August 2021 Accepted: 09 October 2021 Published: 04 November 2021
MSC : 30D35, 35M30, 32W50

This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation

$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $

where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi ^[23], Xu and Cao ^[35], Liu, Cao and Cao ^[17].
- Nevanlinna theory,
- Fermat type,
- entire solution,
- partial differential difference equation
Citation: Wenju Tang, Keyu Zhang, Hongyan Xu. Results on the solutions of several second order mixed type partial differential difference equations[J]. AIMS Mathematics, 2022, 7(2): 1907-1924. doi: 10.3934/math.2022110

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Abstract

This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation

$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $

where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi ^[23], Xu and Cao ^[35], Liu, Cao and Cao ^[17].

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