Application of Fractional Calculus Methods to Viscoelastic Response of Amorphous Shape Memory Polymers

Fang, C-Q.; Sun, H.-Y.; Gu, Jianping

doi:10.1017/jmech.2014.98

Cited by 34 publications

(26 citation statements)

References 37 publications

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“…For more than two centuries, this subject was relevant only in pure mathematics, and Euler, Fourier, Abel, Liouville, Riemann, Hadamard, among others, have studied these new fractional operators, by presenting new definitions and studying their most important properties. However, in the past decades, this subject has proven its applicability in many and different natural situations, such as viscoelasticity [11,26], anomalous diffusion [14,19], stochastic processes [9,29], signal and image processing [31], fractional models and control [24,32], etc. This is a very rich field, and for it we find several definitions for fractional integrals and for fractional derivatives [16,25].…”

Section: Introductionmentioning

confidence: 99%

A Caputo fractional derivative of a function with respect to another function

Almeida

2017

Communications in Nonlinear Science and Numerical Simulation

840

512

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In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.Mathematics Subject Classification 2010: 26A33, 34A08, 65L05, 34K60.

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Section: Introductionmentioning

confidence: 99%

A Caputo fractional derivative of a function with respect to another function

Almeida

2017

Communications in Nonlinear Science and Numerical Simulation

840

512

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“…Note that in general, (10) and (11) are not true for any ∈ L 1 ([a, b]; R). The following results show this fact.…”

Section: Derivative Operators With Respect To a Kernel Functionmentioning

confidence: 99%

“…For more than two centuries, this theory has been treated as a purely theoretical mathematical field, and many mathematicians, such as Liouville, 2 Grünwald, 3 Letnikov, 4 Marchaud,5 and Riemann, 6 have developed this field of research by introducing new definitions and studying their most important properties. However, in the past decades, this subject has proven to be useful in many areas of physics and engineering, such as image processing, 7 fluid mechanics, 8 viscoelasticity, [9][10][11] stochastic processes, 12,13 pollution phenomena, 14 geology, 15 thermal conductivity, 16 and turbulent flows. 17 The classical fractional calculus is based on the well-known Riemann-Liouville fractional integrals…”

Section: Introductionmentioning

confidence: 99%

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

Jleli

Kirane

Samet

2018

Math Methods in App Sciences

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In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved. KEYWORDS boundary value problem, conjugate kernels, fractional calculus, k-derivative x 0 (x − ) −1 ( ) d , > 0 and derivatives of order (D 0 )(x) = ( d dx ) n (I n− 0 )(x), Math Meth Appl Sci. 2019;42:137-160. wileyonlinelibrary.com/journal/mma

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“…Therefore, various forms of fractional-order differential equations are suggested for standard models. In this sense, the fractional-order calculus plays an important role in physics [1], thermodynamics [2], viscoelasticity [3], electrical circuits theory [4], fractances [5], mechatronics systems [6], signal processing [7], chemical mixing [8], chaos theory [9], engineering [10], biological system [11], and other applications [12]. Also, a large number of literatures on the application of fractional-order differential equations (FODEs) in nonlinear dynamics have been improved.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Mathematical Model including Pathogen-Specific Immune System Response with Fractional-Order Differential Equations

Daşbaşı

2018

Computational and Mathematical Methods in Medicine

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In this study, the mathematical model examined the dynamics between pathogen and specific immune system cells (memory T cells) for diseases such as chronic infection and cancer in which nonspecific immune system cells are inadequate to destroy the pathogen and has been suggested by using a system of the fractional-order differential equation with multi-orders. Qualitative analysis of the proposed model reveals the equilibrium points giving important ideas about the proliferation of the pathogen and memory T cells. According to the results of this analysis, the possible scenarios are as follows: the absence of both pathogen and memory T cells, only the existence of pathogen, and the existence of both pathogen and memory T cells. The qualitative analysis of the proposed model has expressed the persistent situations of the disease where the memory T cells either do not be able to respond to the pathogen or continue to exist with the disease-causing pathogen in the host. Results of this analysis are supported by numerical simulations. In the simulations, the time-dependent size of the tumor population under the pressure of the memory T cells was tried to be estimated.

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Application of Fractional Calculus Methods to Viscoelastic Response of Amorphous Shape Memory Polymers

Cited by 34 publications

References 37 publications

A Caputo fractional derivative of a function with respect to another function

A Caputo fractional derivative of a function with respect to another function

A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

Stability Analysis of Mathematical Model including Pathogen-Specific Immune System Response with Fractional-Order Differential Equations

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