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Numerical algorithm to solve the 2D nonlinear multi-term time-fractional Klein–Gordon equation using compact ADI scheme

Author

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  • Kumar, Sonu
  • Srivastava, Nikhil
  • V., Keerthana

Abstract

In this article, a numerical approach is developed to solve the two-dimensional non-linear multi-term time-fractional Klein–Gordon equation (MTTFKGE) using the compact alternating direction implicit (ADI) method. The H3N3-2σ formula is used to discretize the multi-term time-fractional Caputo derivative of order 1<α0,α1,…,αm<2. The ADI method is employed to decompose the two-dimensional system into a sequence of one-dimensional systems along x and y directions. A compact averaging operator is employed in spatial directions, and the nonlinear term is approximated with Taylor series approximation at the super-convergence point σ. The theoretical results on the scheme’s stability and convergence are proved using the energy method in L2 norm. Furthermore, the stability and convergence of the proposed scheme using L2 and L∞ norm errors are validated through numerical examples for various cases of the nonlinear term, mass parameter (k), and variable coefficient in two dimensions.

Suggested Citation

  • Kumar, Sonu & Srivastava, Nikhil & V., Keerthana, 2026. "Numerical algorithm to solve the 2D nonlinear multi-term time-fractional Klein–Gordon equation using compact ADI scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 250(C), pages 1019-1043.
    • Handle: RePEc:eee:matcom:v:250:y:2026:i:c:p:1019-1043
    DOI: 10.1016/j.matcom.2026.07.025
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