mathematics Article Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media Zaheer-ud-Din 1,2, *, Muhammad Ahsan 2,3 , Masood Ahmad 2 , Wajid Khan 2 , Emad E. Mahmoud 4,5 and Abdel-Haleem Abdel-Aty 6,7 1 Department of Basic Sciences, CECOS University of IT and Emerging Sciences Peshawar, Peshawar 25000, Pakistan 2 Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan; ahsankog@uoswabi.edu.pk (M.A.); masood.suf@gmail.com (M.A.); wjdkhan206@gmail.com (W.K.) 3 Department of Mathematics, University of Sawabi, Sawabi 23430, Pakistan 4 Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia; e.mahmoud@tu.edu.sa 5 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt 6 Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia; amabdelaty@ub.edu.sa 7 Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt * Correspondence: zaheeruddin@cecos.edu.pk or zaheer.mth@gmail.com Received: 29 September 2020; Accepted: 11 November 2020; Published: 17 November 2020   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. Keywords: meshless method; integrated MQ RBF; steady-state heat conduction equation 1. Introduction The partial differential equations with nonlocal boundary conditions that emerged in the literature have significant applications in the fields of engineering, astrophysics, and biology. Some of the first nonlocal equations that appeared in the literature are encountered in the field of phase transition and are related to theories due to Van der Waals, Ginzburg, Landau, and Cahn & Hilliard [1]. For instance, such models with nonlocal spatial terms are encountered in the Ohmic heating production [2], in the shear banding formation in metals being deformed under high strain rates [3,4], in the theory of gravitational equilibrium of polytropic stars [5], in the investigation of the fully turbulent behavior of real flows, using invariant measures for the Euler equation [6], in population dynamics [7], Mathematics 2020, 8, 2045; doi:10.3390/math8112045 www.mdpi.com/journal/mathematics