The finite element method is complicated when working with more than 3 space variables and time.
The Kansa Method can be explained by an analogy to a basketball court with many light bulbs suspended all across the ceiling.
E. J. Kansa in very early 1990s made the first attempt to extend radial basis function (RBF), which was then quite popular in scattered data processing and function approximation, to the solution of partial differential equations in the strong-form collocation formulation.
His RBF collocation approach is inherently meshless, easy-to-program, and mathematically very simple to learn.
The most popular RBF in the Kansa method is the multiquadric (MQ), which usually shows spectral accuracy if an appropriate shape parameter is chosen.
The Kansa method approximates the desired function by a linear combination of the RBF in the form: where
Mathematicians prefer the latter for its rigorous and solid theoretical foundation, while engineering users often employ the former since it is easier and simpler and produces the sound results in the majority of cases.
It is known that the FDM is difficult to model an irregular domain for the reason that it usually requires a rectangular grid system.
However, its applications are largely limited by the availability of the fundamental solution of the governing equation.
The driving force behind the scene is that the mesh-based methods such as the standard FEM and BEM require prohibitively computational effort in handling high-dimensional, moving, and complex-shaped boundary problems.
Despite great effort, the rigorous mathematical proof of the solvability of the Kansa method is still missing.
[4][5] propose the symmetric Hermite RBF collocation scheme with sound mathematical analysis of solvability.
However, this strategy requires an additional set of nodes inside or outside of the domain adjacent to the boundary.
The arbitrary placing of these additional nodes gives rise to troublesome issues in the simulation of complex and multiply-connected domain problems.
By using the Green second identity, the modified Kansa method [8][9] is devised to remedy all weaknesses aforementioned.
There exist a number of mathematical theories concerning the family of multiquadric radial basis functions and providing some suggestions on the choice of the shape parameter.
In,[1] the Kansa method is employed to address the parabolic, hyperbolic and elliptic partial differential equations.
Kansa method has recently been extended to various ordinary and PDEs including the bi-phasic and triphasic mixture models of tissue engineering problems,[14][15] 1D nonlinear Burger's equation[16] with shock wave, shallow water equations [17] for tide and current simulation, heat transfer problems,[18] free boundary problems,[19] and fractional diffusion equations.