Hong-Yan Liu, Ji-Huan He and Zheng-Biao Li
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable…
Abstract
Purpose
Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus.
Design/methodology/approach
This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given.
Findings
Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.
Originality/value
Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.
Details
Keywords
Zheng-Biao Li and Wei-Hong Zhu
– The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).
Abstract
Purpose
The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs).
Design/methodology/approach
This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative.
Findings
This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense.
Originality/value
It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.