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This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He's formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values. [ABSTRACT FROM AUTHOR]

In this paper, an equivalent power-form transformation method with a weighted function is applied for solving the one-dimensional fractal Bratu's boundary value equation. Numerical integration solutions obtained from the equivalent Bratu's equation as well as those from its approximate Taylor's series solution reveal that the proposed methodology yields highly accurate solutions. Therefore, it is believed that by applying the power-form transformation, various fractal differential equations in which two scales are needed because of the physical laws involved in modeling the observed phenomena, can be solved by treating the nonlinear terms as equivalent power-form terms. [ABSTRACT FROM AUTHOR]

This article aims to investigate the solution of the fractional Troesch's equation that models the mechanism of current generation in porous electrodes by numerical integration of its equivalent power-form representation and by deriving its approximate Taylor series solution. Simulation results compare to experimental data demonstrate that the overpotential distribution and efficiency within a porous electrode, from the bottom to the top of the bed is predicted by using Troesch's fractal overpotential diffusion equation, when the fractal parameter values are α ≥ 1.2618 for a porous electrode that follows the Hausdorff dimension, or α ≥ 1.236 for El Naschie's fractal space-time. • An equation that models the mechanism of current generation in porous electrodes is investigated. • The equivalent representation power-form expression for Troesch's equation is derived • Two scale dimension transform is used to study fractal Troesch's equation. • The Taylor series method is used to solve the equivalent power-form representation of Troesch's equation. [ABSTRACT FROM AUTHOR]