Fractional diffusion-wave equations on finite interval by Laplace transform

Duan, Jun‐Sheng; Fu, Shouzhong; Wang, Zhong

doi:10.1080/10652469.2013.838759

Cited by 8 publications

(6 citation statements)

References 16 publications

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“…Various methods for the solution of differential equation of fractional order are available in literature, including Laplace method [21,24], Grünwald-Letnikov method [21,25], Adomian method [26] and several others [15,21,[27][28][29][30]. Some attempts to use Mellin Transform and related concepts, have been presented (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Fractional differential equations solved by using Mellin transform

Butera

Paola

2014

Communications in Nonlinear Science and Numerical Simulation

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In this paper, the solution of the multi-order differential equations, by using Mellin transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands

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Section: Introductionmentioning

confidence: 99%

Fractional differential equations solved by using Mellin transform

Butera

Paola

2014

Communications in Nonlinear Science and Numerical Simulation

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“…Laplace transform methods remain a popular method for numerically solving FPDEs, as fractional derivatives are well suited for Laplace transformation [6]. Duan et al [11] also used Laplace transform methods to numerically solve fractional diffusion-wave equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions to a Fractional Diffusion Equation Used in Modelling Dye-Sensitized Solar Cells

2021

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Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement.

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“…For this study, we use the Caputo fractional derivative, which is given by [ 18 ]

where

is the order of the fractional derivative,

denotes the classical derivative of f with respect to its variable of order

and

is the usual Gamma function

…”

Section: Introductionmentioning

confidence: 99%

A Fractional Diffusion Model for Dye-Sensitized Solar Cells

Maldon

Thamwattana

2020

Molecules

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Dye-sensitized solar cells have continued to receive much attention since their introduction by O’Regan and Grätzel in 1991. Modelling charge transfer during the sensitization process is one of several active research areas for the development of dye-sensitized solar cells in order to control and improve their performance and efficiency. Mathematical models for transport of electron density inside nanoporous semiconductors based on diffusion equations have been shown to give good agreement with results observed experimentally. However, the process of charge transfer in dye-sensitized solar cells is complicated and many issues are in need of further investigation, such as the effect of the porous structure of the semiconductor and the recombination of electrons at the interfaces between the semiconductor and electrolyte couple. This paper proposes a new model for electron transport inside the conduction band of a dye-sensitized solar cell comprising of TiO 2 as its nanoporous semiconductor. This model is based on fractional diffusion equations, taking into consideration the random walk network of TiO 2 . Finally, the paper presents numerical solutions of the fractional diffusion model to demonstrate the effect of the fractal geometry of TiO 2 on the fundamental performance parameters of dye-sensitized solar cells, such as the short-circuit current density, open-circuit voltage and efficiency.

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Fractional diffusion-wave equations on finite interval by Laplace transform

Cited by 8 publications

References 16 publications

Fractional differential equations solved by using Mellin transform

Fractional differential equations solved by using Mellin transform

Numerical Solutions to a Fractional Diffusion Equation Used in Modelling Dye-Sensitized Solar Cells

A Fractional Diffusion Model for Dye-Sensitized Solar Cells

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