The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

Abstract

Purpose The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems. Design/methodology/approach In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution. Findings The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models. Originality/value The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

New analytical solution of the fractal anharmonic oscillator using an ancient Chinese algorithm: Investigating how plasma frequency changes with fractal parameter values

“…From Figures 1 and 2, one can conclude that the approximate Here, the blue dashed lines describe the exact numerical solution of equation ( 4) obtained by using the solution derived in Ref. 68., while the color solid lines are the approximate solutions computed from equations ( 5) and (12).…”

“…derived from (12) with α = 1 and Q = 0, with the exact solution of the homogeneous equation ( 4) found in Ref. 68.…”

“…Fractal calculus application to model several physical and engineering phenomena has gained popularity since introducing the two-scale fractal transform that unveils hidden phenomena beyond the conventional continuum mechanics. [1][2][3][4][5][6] The resulting equations of motion are usually described by nonlinear ordinary differential equations whose frequency-amplitude relationships are determined using, for instance, homotopy perturbation methods, [7][8][9][10][11][12] parameter expansion method (PEM), 13 He's frequency formulation, [14][15][16] iteration perturbation method, 17,18 the fractal variational principle, [19][20][21][22][23] energy balance method, 24 energy method, 25 He's frequency formulation based on energy conservation, 26 frequency-amplitude's formula for non-conservative oscillators, 27 the enhanced homotopy perturbation method, 28 Taylor series, 29,30 the equivalent power-form representation, 31 and an ancient Chinese algorithm 32 to name a few.…”

Analytic algorithm for local fractional Caudrey-Dodd-Gibbon-Kaeada equation based on the new iterative method

“…The understanding of eq. ( 16) was discussed in detail by the two-scale fractal theory for various applications [25][26][27][28][29][30][31][32][33][34]. We rewrite the problem (17):…”

A heuristic review on the homotopy perturbation method for non-conservative oscillators

“…The homotopy perturbation method always leads to an approximate solution of a nonlinear problem, but sometimes an exact one can be obtained. 43,44 The method was originally proposed to solving differential equations, but it can be used to solve fractal differential equations, 45,46 fractional differential equations, 47 and integral equations, 48,49 and difference equations. 50 It is extremely effective for inverse problems.…”