About This Special Issue
There are several numerical methods for solving initial boundary value problems such as Finite Difference Method (FDM), Finite Element Method (FEM) usually based on variational or Galerkin approximation, Boundary Element Method (BEM) based on Integral Representation Method (IRM), Smooth Particle Hydrodynamics (SPH), Moving Particle Method (MPM), Vortex Blob Method, Collocation Method and so on. These methods originate from the difference how to discretize the mathematically continuous equations. Each method has its own long and short points. For example, FEM is very flexible to the complex geometry, but mesh-division may invite a serious problem. Since BEM is developed for linear problems, it’s very efficient means of numerical calculation, but it can’t cope with nonlinearity properly. IRM discussed in the special issue is intended to overcome the limit of BEM. In IRM, the fundamental solution is a key to the method, and we usually use the fundamental solution of the linearlized problem. However, this may also invite some difficulties. In case of FEM, the idea using the variational principle is replaced by Galerkin’s method to cope with problems not having the variational principle. Similarly, IRM should also replaced by Generalized Integral representation Method (GIRM) where the fundamental solution is chosen more flexibly.
Generally speaking, physical phenomena are described as boundary value problems in differential equations. We refer to this type of problem as a differential-type boundary value problem. If we use a fundamental solution of the differential equations, we can derive integral representations from the differential-type boundary value problem. If we substitute the boundary conditions into the integral representations, we obtain the integral equations. We can determine the unknown variables by solving the integral equations. The integral representations are equivalent to the differential-type boundary value problem. Hence, we refer to the boundary value problem expressed by the integral representations as the integral-type boundary value problem.
In the FEM, we use simple interpolation functions in the elements. This may reduce the degrees of freedom of the interpolation functions, and we overcome this difficulty by increasing the number of elements. As such, we face a serious problem in the mesh division. In the IRM, since the continuity of unknown variables between the elements is not required explicitly, an easier division into elements and a higher precision interpolation are realized, and a mesh-free approach would be possible.