Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SOLUTIONS FOR QUADRATIC TRINOMIAL PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS IN ℂn

  • Molla Basir Ahamed (Department of Mathematics Jadavpur University) ;
  • Sanju Mandal (Department of Mathematics Jadavpur University)
  • Received : 2023.07.23
  • Accepted : 2024.06.07
  • Published : 2024.09.01
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Abstract

In this paper, we utilize Nevanlinna theory to study the existence and forms of solutions for quadratic trinomial complex partial differential-difference equations of the form aF2 + 2ωFG + bG2 = exp(g), where ab ≠ 0, ω ∈ ℂ with ω2 ≠ 0, ab and g is a polynomial in ℂn. In order to achieve a comprehensive and thorough analysis, we study the characteristics of solutions in two specific cases: one when ω2 ≠ 0, ab and the other when ω = 0. Because polynomials in several complex variables may exhibit periodic behavior, a property that differs from polynomials in single complex variables, our study of finding solutions of equations in ℂn is significant. The main results of the paper improved several known results in ℂn for n ≥ 2. Additionally, the corollaries generalize results of Xu et al. [Rocky Mountain J. Math. 52(6) (2022), 2169-2187] for trinomial equations with arbitrary coefficients in ℂn. Finally, we provide examples that endorse the validity of the conclusions drawn from the main results and their related remarks.

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Acknowledgement

The authors express their gratitude to the referee for the helpful suggestions and insightful comments aimed at improving the exposition of the paper. The first author is supported by the DST FIST (SR/FST/MS-II/2021/101(C)), Department of Mathematics, Jadavpur University. The Second author is supported by CSIR-SRF (File No: 09/0096(12546)/2021-EMR-I, dated: 18/12/2023), Govt. of India, New Delhi.

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