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L3 approximation of Caputo derivative and its application to time-fractional wave equation-(I)

Author

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  • Srivastava, Nikhil
  • Singh, Vineet Kumar

Abstract

In this manuscript, first of all, we have developed two novel numerical approximations (called L3 and ML3 approximation) of the Caputo fractional derivative of order α∈(1,2). We have used the cubic interpolating polynomial on uniform grid points [(tj−2,Uj−2), (tj−1,Uj−1), (tj,Uj), (tj+1,Uj+1)] for 2≤j≤k−1 while the quadratic interpolating polynomial is applied on the first interval [t0,t1]. We have modified the L3 approximation by using cubic Hermite interpolation in the sub-interval [t0,t2]. The novel L3 and ML3, both approximation are second order accurate for all α. Both approximations are tested on various examples and gives highly accurate results. Later, using this L3 approximation, a difference scheme is proposed to solve the time-fractional wave equation (TFWE). The proposed difference scheme is second order accurate in space and time for all α. The scheme is again tested on three numerical problems of TFWE, and the comparative study of the numerical results by the proposed scheme with some existing schemes is also provided to show the effectiveness and accuracy of our scheme.

Suggested Citation

  • Srivastava, Nikhil & Singh, Vineet Kumar, 2023. "L3 approximation of Caputo derivative and its application to time-fractional wave equation-(I)," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 532-557.
    • Handle: RePEc:eee:matcom:v:205:y:2023:i:c:p:532-557
    DOI: 10.1016/j.matcom.2022.10.003
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    References listed on IDEAS

    as
    1. Alikhanov, Anatoly A. & Huang, Chengming, 2021. "A high-order L2 type difference scheme for the time-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    2. Maurya, Rahul Kumar & Devi, Vinita & Srivastava, Nikhil & Singh, Vineet Kumar, 2020. "An efficient and stable Lagrangian matrix approach to Abel integral and integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 374(C).
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    More about this item

    Keywords

    L3 approximation; ML3 approximation; Caputo derivative; Time-fractional wave equation; Finite difference scheme;
    All these keywords.

    JEL classification:

    • L3 - Industrial Organization - - Nonprofit Organizations and Public Enterprise

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