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A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

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  • Hussain, Manzoor
  • Ghafoor, Abdul
  • Hussain, Arshad
  • Haq, Sirajul
  • Ali, Ihteram
  • Arifeen, Shams Ul

Abstract

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

Suggested Citation

  • Hussain, Manzoor & Ghafoor, Abdul & Hussain, Arshad & Haq, Sirajul & Ali, Ihteram & Arifeen, Shams Ul, 2024. "A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 130-157.
    • Handle: RePEc:eee:matcom:v:223:y:2024:i:c:p:130-157
    DOI: 10.1016/j.matcom.2024.04.003
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    References listed on IDEAS

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    1. Sarker, Pratik & Chakravarty, Uttam K, 2020. "A generalization of the method of lines for the numerical solution of coupled, forced vibration of beams," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 170(C), pages 115-142.
    2. Arshad Hussain & Marjan Uddin & Sirajul Haq & Hameed Ullah Jan & Fazlollah Soleymani, 2021. "Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines," Journal of Mathematics, Hindawi, vol. 2021, pages 1-11, December.
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