Meromorphic solutions of Bi-Fermat type partial differential and difference equations

• Published in: Analysis and mathematical physics Vol. 14; no. 6
• Main Authors: Gao, Yingchun, Liu, Kai
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.12.2024
• Subjects: Difference equations; Functional equations; Partial differential equations
• ISSN: 1664-2368; 1664-235X
• DOI: 10.1007/s13324-024-00989-w

Fermat type functional equation with four terms f(z)n+g(z)n+h(z)n+k(z)n=1is difficult to solve completely even if n=2,3, in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation f(z1,z2)2+∂f(z1,z2)∂z12+g(z1,z2)2+∂g(z1,z2)∂z12=1in C2. In addition, we consider the Bi-Fermat type cubic difference equation f(z)3+g(z)3+f(z+c)3+g(z+c)3=1in C and obtain partial meromorphic solutions on the above equation.