Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.
A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension
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AbstractPurpose -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.
We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.
The Fractional Complex Transform is extended to solve exactly time-fractional differential equations with the modified Riemann-Liouville derivative. How to incorporate suitable boundary/initial conditions is also discussed
He?s fractional derivative is adopted in this paper to study the heat
conduction in fractal medium. The fractional complex transformation is
applied to convert the fractional differential equation to ordinary
different equation. Boltzmann transform and wave transform are used to
further simplify the governing equation for some special cases. Silkworm
cocoon are used as an example to elucidate its natural phenomenon.
Time-fractional differential equations can accurately describe heat conduction in fractal media, such as wool fibers, goose down and polar bear hair. The fractional complex transform is used to convert time-fractional heat conduction equations with the modified Riemann-Liouville derivative into ordinary differential equations, and exact solutions can be easily obtained. The solution process is straightforward and concise
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