Integral expressions of solutions to higher order $ \lambda $-weighted Dirac equations valued in the parameter dependent Clifford algebra

Xiaojing Du; Xiaotong Liang; Yonghong Xie; Xiaojing Du; Xiaotong Liang; Yonghong Xie

doi:10.3934/math.2025050

AIMS Mathematics

2025, Volume 10, Issue 1: 1043-1060. doi: 10.3934/math.2025050

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Research article

Integral expressions of solutions to higher order $ \lambda $-weighted Dirac equations valued in the parameter dependent Clifford algebra

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China

Received: 08 November 2024 Revised: 20 December 2024 Accepted: 09 January 2025 Published: 17 January 2025
MSC : 30E20, 35G35, 45E05

First, some important properties of functions valued in the parameter dependent Clifford algebra $ \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) $ were studied. Second, we obtained Cauchy-Pompeiu formulae for functions valued in $ \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) $ and the integral representation of solutions to the higher order $ \lambda $-weighted Dirac equation, respectively. Finally, the integral representation of solutions to bilateral higher order $ \lambda $-weighted Dirac equations was derived.
Citation: Xiaojing Du, Xiaotong Liang, Yonghong Xie. Integral expressions of solutions to higher order $ \lambda $-weighted Dirac equations valued in the parameter dependent Clifford algebra[J]. AIMS Mathematics, 2025, 10(1): 1043-1060. doi: 10.3934/math.2025050

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Abstract

First, some important properties of functions valued in the parameter dependent Clifford algebra $ \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) $ were studied. Second, we obtained Cauchy-Pompeiu formulae for functions valued in $ \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) $ and the integral representation of solutions to the higher order $ \lambda $-weighted Dirac equation, respectively. Finally, the integral representation of solutions to bilateral higher order $ \lambda $-weighted Dirac equations was derived.

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