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Abstract
This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.
Highlights
The growing interest in the theory and applications of fractional calculus has become the motivation for many researchers in recent years
Solving fractional differential equations (FDEs) using numerical methods has been seen as an ongoing research trend
(2021) 2021:147 challenging compared to the ordinary (ODEs) and partial differential equations (PDEs) in general
Summary
The growing interest in the theory and applications of fractional calculus has become the motivation for many researchers in recent years. Fractional calculus has attracted attention of experts from all over the world. Solving fractional differential equations (FDEs) using numerical methods has been seen as an ongoing research trend. The analytical solutions of most FDEs are Sunarto et al Advances in Difference Equations (2021) 2021:147 challenging compared to the ordinary (ODEs) and partial differential equations (PDEs) in general. Numerical solutions are actively being found by proposing new numerical approximation techniques to solve the FDEs. Some notable numerical methods have been developed to solve the fractional partial derivatives problems [1, 2, 14, 19, 20, 29]
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