Quintic non-polynomial spline for time-fractional nonlinear Schr\xf6dinger equation

Qinxu Ding,Patricia J Y Wong

doi:10.1186/s13662-020-03021-0

Qinxu Ding, Patricia J Y Wong

Open Access

https://doi.org/10.1186/s13662-020-03021-0

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Abstract

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and (2-gamma )th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

Highlights

In the past few decades, fractional differential equations have gained much importance due to their usefulness in modeling phenomena in various areas such as physics, engineering, finance, biology and chemistry [12, 33, 46, 51]
To cite some recent developments: in 2019 Jajarmi and Baleanu [26] studied a general form of fractional optimal control problems involving fractional derivative with singular or non-singular kernel; Jothimani et al [30] discussed an exact controllability of nondensely defined nonlinear fractional integrodifferential equations with the Hille–Yosida operator; Valliammal et al [61] studied the existence of mild solutions of fractional-order neutral differential system with statedependent delay in Banach space
In 2020 Jajarmi et al [28] investigated a fractional version of SIRS model for the HRSV disease; Baleanu et al [2] proposed a new fractional model for the human liver involving the Caputo–Fabrizio fractional derivative; Baleanu et al [3] studied the fractional features of a harmonic oscillator with position-dependent mass; Sajjadi et al [54] discussed the chaos control and synchronization of a hyperchaotic model in both the frameworks of classical and of fractional calculus; Jajarmi and Baleanu [27] proposed a new iterative method to generate the approximate solution of nonlinear fractional boundary value problems in the form of uniformly convergent series; Shiri et al Ding and Wong Advances in Difference Equations

Summary

Introduction

In the past few decades, fractional differential equations have gained much importance due to their usefulness in modeling phenomena in various areas such as physics, engineering, finance, biology and chemistry [12, 33, 46, 51]. Motivated by the above research, in this paper we consider the following time-fractional nonlinear Schrödinger equation:. We define the quintic non-polynomial spline as follows. Using the continuity of the first and third derivatives of the spline at x = xj+1, i.e., Pj(,1n)(xj+1) = Pj(+11) ,n(xj+1) and Pj(,3n)(xj+1) = Pj(+31) ,n(xj+1), we obtain the following relations for 1 ≤ j ≤ M – 1:. Remark 2.1 In order to compute the numerical solution Ujn, 1 ≤ j ≤ M – 1, we need another two equations besides (2.7) or (2.9). Theorem 3.1 (Stability) The numerical scheme (2.25)–(2.27) or equivalently (3.1) is unconditionally stable with respect to the initial data. It follows from (3.19) that ρn dn(m) 2. Theorem 4.1 (Solvability) The numerical scheme (2.25)–(2.27) or equivalently (3.1) is uniquely solvable

Convergence of the numerical scheme

Conclusion

Methods