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

Nikhil Srivastava and Vineet Kumar Singh

Mathematics and Computers in Simulation (MATCOM), 2023, vol. 205, issue C, 532-557

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.

Keywords: L3 approximation; ML3 approximation; Caputo derivative; Time-fractional wave equation; Finite difference scheme (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:205:y:2023:i:c:p:532-557

DOI: 10.1016/j.matcom.2022.10.003

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