An Introduction to the Method of Fundamental Solutions

Over the past two decades, the method of fundamental solutions (MFS) has attracted great attention and has been used extensively for the solution of scientific and engineering problems. The MFS is a boundary meshless collocation method which has evolved from the boundary element method. In it, the approximate solution is expressed as a linear combination of fundamental solutions of the operator in the governing partial differential equation. One of the main attractions of the MFS is the simplicity with which it can be applied to the solution of boundary value problems in complex geometries in two and three dimensions. The method is also known by many different names in the literature such as the charge simulation method, the de-singularization method, the virtual boundary element method, etc. Despite its effectiveness, the original version of the MFS is confined to solving boundary value problems governed by homogeneous partial differential equations. To address this limitation, we introduce various types of particular solutions to extend the method to solving general inhomogeneous boundary value problems employing the method of particular solutions. This book consists of two parts. Part I aims to provide theoretical support for beginners. In the spirit of reproducible research and to facilitate the understanding of the method and its implementation, several MATLAB codes have been included in Part II. This book is highly recommended for use by post-graduate researchers and graduate students in scientific computing and engineering.

Alexander Cheng is Emeritus Dean of Engineering and Professor of Civil Engineering at the University of Mississippi. His research covers the boundary element method, method of fundamental solutions, radial basis function collocation method, groundwater flow, saltwater intrusion, poromechanics, and nanomechanics. He has authored or co-authored five books, including one on Trefftz and Collocation Methods, and co-edited four books and eighteen conference proceedings books. He has published more than 180 refereed journal articles. He currently serves as the Editor-in-Chief of Engineering Analysis with Boundary Elements (Elsevier). He was formerly an Associate Editor for several journals, including Transport in Porous Media (Springer) and Journal of Engineering Mechanics (ASCE). He was the recipient of the George Green Medal, the Maurice A Biot Medal, and the Walter L Huber Civil Engineering Research Prize of ASCE.

Ching-Shyang (CS) Chen is currently Professor of Mathematics at the University of Southern Mississippi where he served as the Chair of the Department of Mathematics during 2005–2010. His main research interests lie in meshless methods, and more specifically, in using radial basis functions and the method of fundamental solutions for solving partial differential equations. He is currently serving as an Associate Editor in Engineering Analysis with Boundary Elements and Advances in Applied Mathematics and Mechanics. He has published 170 research papers and co-authored three research monographs. He was a recipient of the George Green Medal and has received a Fulbright Specialists Award provided by the US Department of State and the J William Fulbright Foreign Scholarship Board.

Andreas Karageorghis completed both his undergraduate and graduate studies at the University of Oxford. After holding positions at the University of Kentucky (USA), the University of Wales (UK) and Southern Methodist University (USA), he joined the Department of Mathematics and Statistics of the University of Cyprus where he is currently Professor. His research interests include numerical algorithms and scientific computing, and particularly the method of fundamental solutions. He was an early developer of the MFS starting in the 1980s, and one of the most prolific authors in the subject. He has published over 170 research papers in international journals. He co-organized and hosted the Workshop on Method of Fundamental Solutions in Ayia Napa, Cyprus, 2007, the first such international gathering on the subject. The results were published in the edited book The Method of Fundamental Solutions — A Meshless Method (2008). He is currently serving on the editorial boards of Engineering Analysis with Boundary Elements and Numerical Algorithms.