2025/07/22 by Hong-Yan Xu, Hong Yan Xu, H. M. Srivastava +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Mathematics and Applications #Meromorphic and Entire Functions
paper · doi:10.1017/s0013091525100825
crossref issued 2025/07/22 · crossref published 2025/07/22 · crossref published-online 2025/07/22 · openalex publication_date 2025/07/22 · crossref created 2025/07/22 · openalex created_date 2025/10/10 · crossref published-print 2025/11/01 · crossref deposited 2026/05/20 · crossref indexed 2026/07/30 · openalex updated_date 2026/08/04
Abstract The purpose of this article is to explore the solutions of the following nonlinear partial differential equations: P1(u)2+P2(u)2=eg and P1(u)2+2α P1(u)P2(u)+P2(u)2=eg, where α2∈ ℂ∖\0,1\ , g ( z ) is a polynomial, aj,bj,cj(j=1,2) are constants in ℂ , and P1(u)=a1 u+b1 uz1+c1uz2 and \mathcal P2(u)=a2 u+b2uz1+c2uz2. 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.