Meshless solutions to multi-dimensional integral and partial differential equations

Abstract

Many important problems in physics, quantum chemistry, biology, economics, etc. are expressed as multi-dimensional (MD) integral equations (IEs) and partial differential equations (PDEs) that are difficult to solve with the dominant numerical techniques such as finite elements, difference, and volume methods. The main problem with multi-dimensional problems is the curse of dimensionality requiring increasingly more computer memory and speed. A radial basis function (RBF) method is recommended that possesses exponential convergence as the key technique to circumvent the curse of dimensionality combined with other techniques to solve MD problems numerically.