Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

Abstract

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

“…Nonlocal BCs are widely used for modeling various problems in science and engineering. Recent applications include their use with reaction-diffusion equations to model heat conduction in bioreactors [24], as well as in anisotropic and inhomogeneous media [25]. This type of BCs is commonly used to describe the relationship between the solution values at multiple points, so the BCs contain the integral expression of the solution.…”

“…While Pao [28,29] investigated the existence and asymptotic behavior of the solution to a general reaction-diffusion equation with nonlocal BCs, analytical solution techniques have not been sufficiently investigated, even for linear equations. The majority of the results in the literature involve the development of numerical methods, for which exact solutions for simple cases are used to benchmark the accuracy of numerical solutions [25,27,[30][31][32][33][34][35]. Recent numerical methods also involve the use of neural networks [36].…”