There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The development of the method of fundamental solutions (MFS) and related methods for the numerical solution of scattering and radiation problems in fluids and solids is described and reviewed A brief review of the developments and applications in all areas of the MFS over the last five years is also given Future possible areas of applications in fields related to scattering and radiation problems are identified

https://inside.mines.edu/~pamartin/ref-paps/R080_EABEw.pdf

The method of fundamental solutions for scattering and radiation problems

Radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable and appear to have certain advantages over the traditional coordinates-based functions. In contrast to the traditional meshed-based methods, the RBF collocation methods are mathematically simple and truly meshless, which avoid troublesome mesh generation for high-dimensional problems involving irregular or moving boundary. This opening chapter begins with the introduction to RBF history and its applications in numerical solution of partial differential equations and then gives a general overview of the book.

Recent Advances in Radial Basis Function Collocation Methods

https://msvlab.hre.ntou.edu.tw/cite/JCP-Young-2005.pdf

Novel meshless method for solving the potential problems with arbitrary domain

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

/pdf/linear-subsets-of-nonlinear-sets-in-topological-vector-3amm15jpk9.pdf

Linear subsets of nonlinear sets in topological vector spaces

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. The basic ideas of the MFS were introduced by Kupradze and Alexidze and its modern form was proposed by Mathon and Johnston. In this work, we investigate certain aspects of a particular version of the MFS, also known as the Charge Simulation Method, when it is applied to the Dirichlet problem for Laplace's equation in a disk.

Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems

Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation

In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration. In our study, the singularities of the fundamental solutions lie on a prescribed pseudo―boundary ― the boundary of a domain which embraces the domain of the problem under consideration. We extend previous density results of Kupradze and Aleksidze, and of Bogomolny, to more general domains and partial differential operators, and with respect to more appropriate norms. Our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement, namely the segment condition. Our density results are with respect to the norms of the spaces C l (Ω). Analogous density results are obtainable with respect to Holder norms. We have studied approximation by fundamental solutions of the Laplacian, biharmonic and m―harmonic and modified Helmholtz and poly-Helmholtz operators. In the case of elliptic systems, we obtain analogous density results for the Cauchy―Navier operator as well as for an operator which arises in the linear theory of thermo―elasticity. We also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space. Finally, we show that linear combinations of fundamental solutions of operators of order m > 4, with singularities lying on a prescribed pseudo-boundary, are not in general dense in the corresponding solution space.

/pdf/applicability-and-applications-of-the-method-of-fundamental-o8so14py69.pdf

Applicability and applications of the method of fundamental solutions

A matrix decomposition RBF algorithm: Approximation of functions and their derivatives

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. Finally, we test the algorithm numerically.

/pdf/numerical-analysis-of-the-mfs-for-certain-harmonic-problems-2eqkfdye3j.pdf

Yiorgos-Sokratis Smyrlis

Papers

Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems

Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation

Applicability and applications of the method of fundamental solutions

A matrix decomposition RBF algorithm: Approximation of functions and their derivatives

Numerical analysis of the MFS for certain harmonic problems