Boundary particle method for Laplace transformed time fractional diffusion equations

Fu, Zhuojia; Yang, Haitian

doi:10.1016/j.jcp.2012.10.018

Cited by 226 publications

(86 citation statements)

References 39 publications

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“…It is indeed a memory-intensive and path-dependent phenomenon which might be mathematically represented by using the concept of fractional calculus [24][25][26] . By using the fractional calculus theory [27][28][29], Yin et al [25,26] successfully proposed a framework for modelling strain hardening and softening of geomaterials. The model could be easily incorporated in engineering-oriented finite element method due to its explicit expression.…”

Section: Introductionmentioning

confidence: 99%

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

et al. 2015

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Many constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one potential instrument for modelling memory-dependent phenomena. In this paper, the physical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail. Then a modified elastoplastic constitutive model is proposed for evaluating the long-term deformation of granular soils under cyclic loading by incorporating the concept of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow potential considering particle breakage is used. Test results of several types of granular soils are used to validate the model performance. Abstract Lots of constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one potential instrument for modelling memory-dependent phenomena. In this paper, the physical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail. Then a modified elasto-plastic constitutive model is proposed for evaluating the long-term deformation of granular soils under cyclic loading by incorporating the concept of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow potential considering particle breakage is used. Test results of several types of granular soils are used to validate the model performance.

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Section: Introductionmentioning

confidence: 99%

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

et al. 2015

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“…We will also investigate the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛o n the generalized weighted Nikolskii-Morrey spaces, see for example, [30]. These results may be applicable to some problems of partial differential equations; see for example [6,7,19,20,26,28,30,43].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Akbulut

Hasanov

2016

Open Mathematics

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Abstract:In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛, which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces M p;' .w/. We find the sufficient conditions on the pair .' 1 ; ' 2 / with w 2 A p;q which ensures the boundedness of the operators T ;˛a re given in terms of Zygmund-type integral inequalities on .' 1 ; ' 2 / and w, which do not assume any assumption on monotonicity of ' 1 .x; r/; ' 2 .x; r/ in r.

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“…It is worth mentioning that fractional calculus deals with differential operators of arbitrary order, being an important tool to describe memory and hereditary effects of properties of many materials and processes [15]. Fractional calculus also presents a broad range of applications in process systems engineering, rheology, viscoelasticity, acoustics, optics, chemical physics, robotics, electrical engineering, bioengineering, anomalous diffusion [16][17][18][19][20][21]. The solution of the model here proposed is a nonlinear algebraic equation based on the Mittag-Leffer function, which according to the value of parameter α can turn into, for example, an exponential function.…”

Section: Introductionmentioning

confidence: 99%

Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis

2018

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Abstract:A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of order α and an additional term multiplied by a parameter k. Using a hybrid parameter estimation scheme (genetic algorithm and a simplex-based algorithm), the model parameters were estimated as k = 3.42 ± 0.12 and α = 1.196 ± 0.027, while a correlation coefficient value of 0.997 was obtained. For the sake of comparison, parameter α was set equal to 1 and an integer order model was also studied, resulting in a one-parameter model with k = 3.11 ± 0.28. Joint confidence regions are calculated for the fractional order model, showing that the derivative order is statistically different from 1. Finally, an independent validation sample of color component B equal to 96 obtained from a sample with olive oil mass fraction equal to 0.25 is used for prediction purposes. The fractional model predicted the color B value equal to 93.1 ± 6.6.

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Boundary particle method for Laplace transformed time fractional diffusion equations

Cited by 226 publications

References 39 publications

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis

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