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Mapping techniques for collocation method of time-fractional convection–diffusion equations in domains with cracks

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  • Jang, Bongsoo
  • Kim, Hyunju

Abstract

This paper proposes numerical methods that effectively deal with time-fractional convection–diffusion equations containing crack singularities. To deal with singularities, we design the geometrical mapping whose push-forward from the parameter space into the physical space generates point singularity functions based on the parametrization of the circular arc and NURBS (non-uniform rational B-spline). We adopt the collocation method with B-spline basis functions to approximate the solution in the spatial direction and enrich the approximation space by k-refinements in IGA (Isogeometric Analysis). For the discretization along the temporal direction, we employ the explicit Predictor-Corrector (PC) scheme that has the order 2−ν and 3−ν of the truncation error for the linear and quadratic interpolation, respectively. Taking advantage of the NURBS geometrical mapping, we demonstrate the performance of the proposed methods applying to time-fractional convection–diffusion equations with nonlinear terms on curved domains containing crack singularities.

Suggested Citation

  • Jang, Bongsoo & Kim, Hyunju, 2024. "Mapping techniques for collocation method of time-fractional convection–diffusion equations in domains with cracks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 60-79.
  • Handle: RePEc:eee:matcom:v:217:y:2024:i:c:p:60-79
    DOI: 10.1016/j.matcom.2023.10.014
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    References listed on IDEAS

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    1. Zhang, Juan & Zhang, Xindong & Yang, Bohui, 2018. "An approximation scheme for the time fractional convection–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 305-312.
    2. Kim, Hyunju & Kim, Keon Ho & Lee, Seyeon & Jang, Bongsoo, 2020. "New explicit and accelerated techniques for solving fractional order differential equations," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    3. Wu, Longyuan & Zhai, Shuying, 2020. "A new high order ADI numerical difference formula for time-fractional convection-diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    4. Zhang, Jun & Wang, JinRong, 2018. "Numerical analysis for Navier–Stokes equations with time fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 481-489.
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