English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Fractional derivative relaxation type equations (FREs) including fractionaldiffusion equation and fractional relaxation equation, have been widely used todescribe anomalous phenomena in physics. To utilize the characteristics offractional dynamic systems, this paper proposes a scale-dependent finitedifference method (S-FDM) in which the non-uniform mesh depends on the timefractional derivative order of FRE. The purpose is to establish a stablenumerical method with low computation cost for FREs by making a bridge betweenthe fractional derivative order and space-time discretization steps. Theproposed method is proved to be unconditional stable with (2-{\alpha})-thconvergence rate. Moreover, three examples are carried out to make a comparisonamong the uniform difference method, common non-uniform method and S-FDM interm of accuracy, convergence rate and computational costs. It has beenconfirmed that the S-FDM method owns obvious advantages in computationalefficiency compared with uniform mesh method, especially for long-time rangecomputation (e.g. the CPU time of S-FDM is ~1/400 of uniform mesh method withbetter relative error for time T=500 and fractional derivative orderalpha=0.4).

A scale-dependent finite difference method for time fractional derivative relaxation type equations