A reliable implicit difference scheme for treatments of fourth-order fractional sub-diffusion equation

Abstract

KEYWORDSAbstract. In this paper, a reliable implicit di erence scheme is proposed to analyze the fractional fourth-order subdi usion equation on a bounded domain. The time-fractional derivative operator is characterized in the Ji Huan He's sense, and the space derivative is approximated by the ve-point centered formula. The numerical parameters, i.e. consistency, stability, and convergence analyses of the considered scheme, are proven.

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

“…In literature, eq. 29was called as He's fractional derivative [31][32][33][34][35][36][37][38][39][40][41][42], and it has been applied to biomechanics [38], nanoscale thermodynamics [39], Zakharov-Kuznetsov equation [31], KP-BBM equation [32], solitary theory [33], non-linear vibration [34], coast protection [35,36], high-order sub-diffusion broblem [37], fractional optimal control problems [38], drug release [39][40][41], and biomaterials [42].…”

A fractional model and its application to heat prevention coating with cocoon-like hierarchy

“…There are many definitions of fractional derivatives, this paper adopts the variational iteration algorithm-based definition, and it is called as He's fractional derivative [22] or the fractional derivative in Ji Huan He's sense [23] in the literature. The VIM was first used to solve fractional differential equations in [24], and it has been proved to be effective, easy, and accurate to solve a lot of non-linear differential problems with the approximate values converging rapidly to the exact solutions.…”

Amplitude–frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion

“…There are various definitions on fractional derivatives. [6][7][8][9][10][11][12] This paper uses He's fractional derivative, [13][14][15][16] which is defined through the variational iteration algorithm. 17 We consider the following linear equation of nth order u ðnÞ ðtÞ ¼ fðtÞ…”