The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic-hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.
Reconstruction of the electric field of the Helmholtz equation in 3D
Ton Duc Thang University（胡志明市屯德胜大学）；Gran Sasso Science Institute（格兰萨索科学研究所）；Goethe University Frankfurt（法兰克福歌德大学）；The University of New South Wales（新南威尔士大学）
A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation
University of Science and Technology of China（中国科学技术大学）；Suzhou Institute for Advanced Research, University of Science and Technology of China（中国科学技术大学苏州高等研究院）；Suzhou Big Data & AI Research and Engineering Center（苏州大数据与人工智能研究与工程中心）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
