A symmetric integrated radial basis function method for solving differential equations

Abstract

In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation‐based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high‐order accuracy can be achieved together. Cartesian‐grid‐based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary‐value problems governed by the Poisson and convection‐diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations.

Inverse differential quadrature method: mathematical formulation and error analysis

“…where P e is the Peclet number. The exact solution for the given partial differential equation is given in [55] as ufalse(x,yfalse)=efalse(Pex/2false) sin⁡false(πyfalse)(2e((−Pe)/2) sinh⁡(σx)+sinh⁡(σ(1−x))sinh⁡(σ)), where σ=π2+Pe2/4.…”

“…Consider a steady-state convection-diffusion equation with boundary conditions as follows, where P e is the Peclet number. The exact solution for the given partial differential equation is given in [55] as u(x, y) = e (P e x/2) sin(π y) 2e ((−P e )/2) sinh(σ…”

New approximations for one-dimensional 3-point and two-dimensional 5-point compact integrated RBF stencils

“…One effective way to overcome accuracy reductions is to develop compact local RBF stencils, where the RBF approximations are expressed in terms of nodal values of not only the field variable but also its derivatives [22,23,24,25]. They can be implemented with differentiated RBFs or integrated RBFs on Cartesian grids [24,25,26] or unstructured nodes [23,27]. Nodal derivative values can be included by using the Hermite interpolation approach [23,27] or by means of integration constants [26].…”

“…They can be implemented with differentiated RBFs or integrated RBFs on Cartesian grids [24,25,26] or unstructured nodes [23,27]. Nodal derivative values can be included by using the Hermite interpolation approach [23,27] or by means of integration constants [26]. An advantage of the former over the latter is that its interpolation matrix is symmetric and guaranteed to be invertible.…”

A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

“…This method was first proposed by Kansa [1] for surface approximation and solutions of partial differential equations (PDEs). In later years, RBFs method has attracted a large number of researchers and practitioners to solve many practical problems [2–14]. RBFs based methods have also been numerically tested for solution of Black–Scholes model and its variants [4, 15–17].…”