Multi-Dimensional Analytic Functions for Laplace Equations and Generalized Cauchy–Riemann Equations

Chein-Shan Liu, Zhuojia Fu and Chung-Lun Kuo ()
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Chein-Shan Liu: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Zhuojia Fu: Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
Chung-Lun Kuo: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan

Mathematics, 2025, vol. 13, issue 8, 1-24

Abstract: A new concept of projective solution is introduced for the multi-dimensional Laplace equations. We project the field point onto a characteristic vector to obtain a projective variable, which can be used to reduce the Laplace equations to a second-order ordinary differential equation with only a leading term multiplied by the squared norm of the characteristic vector. The projective solutions involve characteristic vectors as parameters, which must be complex numbers to satisfy a null equation. Since the projective variable is a complex variable, we can construct the analytic function based on the conventional complex analytic function theory. Both the analytic function and the Cauchy–Riemann equations are generalized for the multi-dimensional Laplace equations. A powerful numerical technique to solve the 3D Laplace equation with high accuracy is available by further developing the Trefftz-type bases. Numerical experiments confirm the accuracy and efficiency of the projective solutions method (PSM).

Keywords: Laplace equations; characteristic vector; projective solutions method; analytic functions; generalized Cauchy–Riemann equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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