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An Efficient Local Formulation for Time–Dependent PDEs

Author

Listed:
  • Imtiaz Ahmad

    (Department of Mathematics, University of Swabi, Swabi 23430, Pakistan)

  • Muhammad Ahsan

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

  • Zaheer-ud Din

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

  • Ahmad Masood

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

  • Poom Kumam

    (KMUTT-Fixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
    KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand)

Abstract

In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains.

Suggested Citation

  • Imtiaz Ahmad & Muhammad Ahsan & Zaheer-ud Din & Ahmad Masood & Poom Kumam, 2019. "An Efficient Local Formulation for Time–Dependent PDEs," Mathematics, MDPI, vol. 7(3), pages 1-18, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:3:p:216-:d:209123
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    References listed on IDEAS

    as
    1. Assas, Laila M.B., 2008. "Variational iteration method for solving coupled-KdV equations," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1225-1228.
    2. 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.
    3. Helal, M.A. & Mehanna, M.S., 2007. "A comparative study between two different methods for solving the general Korteweg–de Vries equation (GKdV)," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 725-739.
    4. Moghimi, Mahdi & Hejazi, Fatemeh S.A., 2007. "Variational iteration method for solving generalized Burger–Fisher and Burger equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1756-1761.
    Full references (including those not matched with items on IDEAS)

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