A well-conditioned and efficient implementation of dual reciprocity method for Poisson equation

Suliman Khan; M. Riaz Khan; Aisha M. Alqahtani; Hasrat Hussain Shah; Alibek Issakhov; Qayyum Shah; M. A. EI-Shorbagy; Suliman Khan; M. Riaz Khan; Aisha M. Alqahtani; Hasrat Hussain Shah; Alibek Issakhov; Qayyum Shah; M. A. EI-Shorbagy

doi:10.3934/math.2021724

AIMS Mathematics

2021, Volume 6, Issue 11: 12560-12582. doi: 10.3934/math.2021724

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Research article

A well-conditioned and efficient implementation of dual reciprocity method for Poisson equation

1.
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
2.
LSEC and ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
4.
Department of Mathematical Sciences, Balochistan University of Information Technology Engineering and Management Sciences, Quetta, Pakistan
5.
Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, 050040, Almaty, Kazakhstan
6.
Department of Mathematics and Cybernetics, Kazakh British Technical University, 050000, Almaty, Kazakhstan
7.
Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan
8.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudia Arabia
9.
Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt

Received: 16 April 2021 Accepted: 19 August 2021 Published: 31 August 2021
MSC : 41A05, 42A82, 65F50, 65L10, 65N12, 65N15, 65N38

One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.
- dual reciprocity method,
- multiquadrics,
- compactly supported radial basis functions,
- stability analysis,
- condition number,
- Poisson equation
Citation: Suliman Khan, M. Riaz Khan, Aisha M. Alqahtani, Hasrat Hussain Shah, Alibek Issakhov, Qayyum Shah, M. A. EI-Shorbagy. A well-conditioned and efficient implementation of dual reciprocity method for Poisson equation[J]. AIMS Mathematics, 2021, 6(11): 12560-12582. doi: 10.3934/math.2021724

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Abstract

One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.

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