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Mathematics
Volume 14
Issue 12
10.3390/math14122109
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Article

Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation

by
Kwesi Acheampong
1,
Zhiyun Yu
2 and
Huiqing Zhu
1,*
1
School of Mathematics and Natural Sciences, The University of Southern Mississippi, Hattiesburg, MS 39406, USA
2
School of Mathematics and Information Science, Zhongyuan University of Technology, Zhengzhou 450007, China
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(12), 2109; https://doi.org/10.3390/math14122109 (registering DOI)
Submission received: 27 February 2026 / Revised: 6 June 2026 / Accepted: 9 June 2026 / Published: 12 June 2026
(This article belongs to the Special Issue Mathematical Methods in Machine Learning, Neural Networks and Computer Vision)
Versions Notes

Abstract

This paper proposes a localized Hermite method of approximate particular solutions (LHMAPS) for solving the 2D inhomogeneous Helmholtz-type equations. Building on the local scheme of the localized method of approximate particular solutions (LMAPS) for the Helmholtz-type differential operator, LHMAPS employs Hermite-type local approximations involving both the solution values and their Laplacian to improve the accuracy of LMAPS. The polyharmonic spline (PS) radial basis functions and polynomial basis functions are considered in the formulation of LHMAPS. Numerical experiments are presented to demonstrate the enhanced accuracy achieved by employing Hermite-type local approximations.
Keywords: radial basis functions; method of approximate particular solutions; Hermite interpolation radial basis functions; method of approximate particular solutions; Hermite interpolation

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MDPI and ACS Style

Acheampong, K.; Yu, Z.; Zhu, H. Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation. Mathematics 2026, 14, 2109. https://doi.org/10.3390/math14122109

AMA Style

Acheampong K, Yu Z, Zhu H. Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation. Mathematics. 2026; 14(12):2109. https://doi.org/10.3390/math14122109

Chicago/Turabian Style

Acheampong, Kwesi, Zhiyun Yu, and Huiqing Zhu. 2026. "Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation" Mathematics 14, no. 12: 2109. https://doi.org/10.3390/math14122109

APA Style

Acheampong, K., Yu, Z., & Zhu, H. (2026). Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation. Mathematics, 14(12), 2109. https://doi.org/10.3390/math14122109

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