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

Author

Listed:
  • Zaheer-ud-Din

    (Department of Basic Sciences, CECOS University of IT and Emerging Sciences Peshawar, Peshawar 25000, Pakistan
    Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan)

  • Muhammad Ahsan

    (Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan
    Department of Mathematics, University of Sawabi, Sawabi 23430, Pakistan)

  • Masood Ahmad

    (Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan)

  • Wajid Khan

    (Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan)

  • Emad E. Mahmoud

    (Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
    Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt)

  • Abdel-Haleem Abdel-Aty

    (Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
    Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt)

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.

Suggested Citation

  • Zaheer-ud-Din & Muhammad Ahsan & Masood Ahmad & Wajid Khan & Emad E. Mahmoud & Abdel-Haleem Abdel-Aty, 2020. "Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media," Mathematics, MDPI, vol. 8(11), pages 1-19, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2045-:d:446225
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    References listed on IDEAS

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    1. Kadalbajoo, Mohan K. & Kumar, Alpesh & Tripathi, Lok Pati, 2015. "A radial basis functions based finite differences method for wave equation with an integral condition," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 8-16.
    2. Ballestra, Luca Vincenzo & Pacelli, Graziella, 2013. "Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1142-1167.
    3. Zaheer-ud-Din, & Siraj-ul-Islam,, 2018. "Meshless methods for one-dimensional oscillatory Fredholm integral equations," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 156-173.
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