The closed-form particular solutions for the Laplace operator using oscillatory radial basis functions in 2D

Improving Numerical Accuracy of the Localized Oscillatory Radial Basis Functions Collocation Method for Solving Elliptic Partial Differential Equations in 2D

“…where ||.|| is the Euclidean norm, ε is the shape parameter, J d−1/2 signifies the J Bessel function of the first kind of order d − 1/2, and φ d is called the ORBF of order d. To solve the elliptic PDEs (1) and (2) using the proposed scheme, we require the particular solutions for ∆Φ d = φ d , derived in [8], as follows:…”

“…Φ d k (ξ). Details on deriving the particular solutions using ORBFs can be found in [8]. Next, we choose n q polynomials p s (ξ) = x m−s 1…”

“…Among many other particular solution-based methods, the MAPS requires particular solutions of the corresponding RBFs. To that end, Lamichhane et al [8] derived the particular solutions of Poisson's equation in 2D using ORBFs and successfully implemented the derived particular solutions in the method of approximate particular solutions for solving various elliptic partial differential equations (PDEs). This method is known as the ORBFs collocation method, which outperformed the multiquadric RBF in terms of accuracy.…”