The Kansa method (also known as the unsymmetric radial basis function collocation method) is a meshless numerical method for solving partial differential equations (PDEs) introduced by E. J. Kansa in the early 1990s. The method uses radial basis functions (RBFs) to approximate the solution in a strong-form collocation framework. The method is a domain-type numerical technique in the sense that the problem is discretized not only on the boundary to satisfy boundary conditions but also inside domain to satisfy governing equation.
Formulation Let Ω ⊆ R d {\displaystyle \Omega \subseteq \mathbb {R} ^{d}} be a d {\displaystyle d} -dimensional physical domain and consider the following boundary value problem (BVP)
L u ( x ) = f ( x ) , x ∈ Ω , ( 1 ) u ( x ) = g ( x ) , x ∈ ∂ Ω D , ( 2 ) ∂ u ( x ) ∂ n = h ( x ) , x ∈ ∂ Ω N , ( 3 ) {\displaystyle {\begin{aligned}Lu(x)&=f(x),\quad x\in \Omega ,&&(1)\\[4pt]u(x)&=g(x),\quad x\in \partial \Omega _{D},&&(2)\\[4pt]{\frac {\partial u(x)}{\partial n}}&=h(x),\quad x\in \partial \Omega _{N},&&(3)\end{aligned}}}
where L {\displaystyle L} represents a differential operator and d {\displaystyle d} is the dimensionality of the problem, ∂ Ω D , ∂ Ω N {\displaystyle \partial \Omega _{D},\,\partial \Omega _{N}} denote the Dirichlet and Neumann boundaries, respectively, and ∂ Ω D ∪ ∂ Ω N = ∂ Ω {\displaystyle \partial \Omega _{D}\cup \partial \Omega _{N}=\partial \Omega } . The Kansa method approximates the desired function by a linear combination of the RBF in the form:
u ( x ) ≈ u ( x ) ∗ = ∑ i = 1 N α i φ ( r i ) , ( 4 ) {\displaystyle u(x)\approx u(x)^{*}=\sum _{i=1}^{N}\alpha _{i}\varphi (r_{i}),\qquad (4)}
where α i {\displaystyle \alpha _{i}} are the coefficients to be determined, φ ( r i ) {\displaystyle \varphi (r_{i})} denotes the RBF such as the multiquadric (MQ), and r i = ‖ x − x i ‖ {\displaystyle r_{i}=\|x-x_{i}\|} , where the x i {\displaystyle x_{i}} are the centres of the RBFs. In order to guarantee the uniqueness of the solution, a polynomial term may be added as follows:
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