10.1007/s40096-022-00506-9

Stability and convergence of a new hybrid method for fractional partial differential equations

  • Kokab Chalambari1,
  • Hamideh Ebrahimi*1,
  • Zeinab Ayati2,
  1. Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, IR
  2. Department of Engineering Sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, Rudsar-Vajargah, IR

Published in Issue 2023-01-10

How to Cite

Chalambari, K., Ebrahimi, H., & Ayati, Z. (2023). Stability and convergence of a new hybrid method for fractional partial differential equations. Mathematical Sciences, 18(3 (September 2024). https://doi.org/10.1007/s40096-022-00506-9
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Abstract

Abstract Radial basis functions (RBFs) have significant role in approximating scattered data. The RBFs with other numerical methods prevent the deficiencies of RBFs. For instance, the radial basis function finite difference method provides geometric flexibility, but the resulting matrix of coefficients is ill-conditioned, to overcome this problem, we can use the QR factorization method for the matrix of coefficients. That leads to the idea of RBF-QR method . The matrix of coefficients is well-conditioned, and the solution is stable when the shape parameter tends to zero. In this article, the RBF-QR method applied to find the approximate solution of constant and variable orders fractional differential equations in the Caputo fractional derivative sense. Gaussian RBF has been involved in this method, and it is converted to Chebyshev polynomials for time and space discretization. The stability and convergence of the method are established, and numerical results are presented to show the accuracy and efficiency of the proposed method.

Keywords

  • Radial basis function,
  • RBF-QR method,
  • Variable-order fractional differential equations,
  • Chebyshev polynomials,
  • Shape parameter,
  • Caputo fractional derivative,
  • Hypergeometric function

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