Kansa method

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.

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. 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{align} where L represents a differential operator and d is the dimensionality of the problem, \partial \Omega_D,\, \partial \Omega_N denote the Dirichlet and Neumann boundaries, respectively, and \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)\approx u(x)^{*} = \sum_{i=1}^N \alpha_i\varphi (r_i),\qquad(4) where \alpha_i are the coefficients to be determined, \varphi(r_i) denotes the RBF such as the multiquadric (MQ), and r_i = \|x - x_i\| , where the 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: u(x)^{*} = \sum_{i=1}^N \alpha_i\varphi(r_i) + \sum_{k=1}^M \alpha_{k+N} \gamma_k(x),\qquad (5) where \gamma_k(x) is the polynomial. The RBF interpolation (4) and (5) are both often used in practice. 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. Substituting Eq. (4) or (5) into Eqs. (1–3) yields the resulting algebraic equation system: \mathbf{A}\alpha = b,\qquad (6) where \mathbf{A} = \left(\begin{matrix} L(\varphi) & L(\gamma) \\[5pt] \varphi & \gamma \\[8pt] \dfrac{\partial\varphi}{\partial n} & \dfrac{\partial\gamma}{\partial n}\\[8pt] \gamma &0 \end{matrix} \right),\quad \mathbf{b}=\left(\begin{matrix} f\\g\\h\\0 \end{matrix} \right),\quad \varphi = \varphi(x_i,x_j),\quad \gamma = \gamma_k(X_i). \qquad (7) After expansion coefficients \alpha_i are evaluated, the desired function can be calculated from Eq. (4) or (5).

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