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A study of transcendental entire solutions of several nonlinear partial differential equations

Part of: Difference and functional equations Difference equations Entire and meromorphic functions, and related topics Equations and systems of special type

Published online by Cambridge University Press:  22 July 2025

Hong Yan Xu  and
Hari Mohan Srivastava
Hong Yan Xu
Affiliation:
School of Mathematics and Physics, Suqian University, Suqian, Jiangsu, P. R. China School of Mathematics and Computer Science, Shangrao Normal University, Shangrao Jiangxi, P. R. China (xhyhhh@126.com)
Hari Mohan Srivastava*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea Department of Mathematics and Informatics, Azerbaijan University, Baku, Azerbaijan Section of Mathematics, International Telematic University Uninettuno, Rome, Italy Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City, Taiwan
*
Corresponding author: Hari Mohan Srivastava, email: harimsri@math.uvic.ca
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Abstract

The purpose of this article is to explore the solutions of the following nonlinear partial differential equations:

\begin{equation*}\mathcal{P}_1(u)^2+\mathcal{P}_2(u)^2=e^{g}\end{equation*}

and

\begin{equation*}\mathcal{P}_1(u)^2+2\alpha \mathcal{P}_1(u)\mathcal{P}_2(u)+\mathcal{P}_2(u)^2=e^{g},\end{equation*}

where $\alpha^2\in \mathbb{C}\setminus\{0,1\}$, g(z) is a polynomial, $a_j,b_j,c_j(j=1,2)$ are constants in $\mathbb{C}$, and

\begin{equation*}\mathcal{P}_1(u)=a_1 u+b_1 u_{z_1}+c_1u_{z_2}\quad \text{and} \quad \mathcal {P}_2(u)=a_2 u+b_2u_{z_1}+c_2u_{z_2}.\end{equation*}

The description of the existence conditions and the forms of the solutions for the above partial differential equations demonstrate that our results improve and generalise the previous results given by Saleeby, Cao and Xu. Moreover, some of our examples corresponding to every case in our theorems reveal the significant difference in the order of solutions for equations from a single variable to several variables.

Keywords

transcendental and meromorphic solutionsentire solutionspartial differential equationsNevanlinna value distribution theory

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 35M30: Systems of mixed type 39A45: Equations in the complex domain

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 41
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

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