<abstract><p>The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.</p></abstract>
The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to O(τ2−α+h2) under strong regularity and O(τα+h2) under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.
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