Caputo-type modification of the Hadamard fractional derivatives
10 Aug 2012-Advances in Difference Equations (Springer International Publishing)-Vol. 2012, Iss: 1, pp 142
TL;DR: In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced, and the properties of the modified derivatives are studied, including their properties of memory effect.
Abstract: Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.
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TL;DR: A numerical method, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented and the efficiency and applicability of the method are shown.
617 citations
Cites methods from "Caputo-type modification of the Had..."
...For some special cases of ψ, we obtain the Caputo fractional derivative [25], the Caputo–Hadamard fractional derivative [12, 15] and the Caputo–Erdélyi–Kober fractional derivative [18]....
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TL;DR: In this paper, the authors introduce new fractional integration and differentiation operators based on the standard fractional calculus iteration procedure on conformable derivatives and define spaces and present some theorems related to these operators.
Abstract: This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.
300 citations
TL;DR: In this paper, the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana-Baleanu fractional derivative were discussed. And the stability of such equations in the sense of Ulam was studied.
Abstract: In this paper, we discuss the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana–Baleanu fractional derivative. Benefiting from the Gronwall inequality in the frame of Riemann–Liouville fractional integral, we establish a Gronwall inequality in the frame of Atangana–Baleanu fractional integral. Then, we study the stability of such equations in the sense of Ulam.
248 citations
TL;DR: In this article, the generalized fractional derivative on ACγ [a, b] is defined, where γn−1f ∈ AC[a,b], where δ = x 1−ρ d dx.
Abstract: In this manuscript, we define the generalized fractional derivative on ACγ [a,b], the space of functions defined on [a,b] such that γn−1f ∈ AC[a,b], where γ = x1−ρ d dx . We present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version. c ©2017 All rights reserved.
191 citations
Cites background from "Caputo-type modification of the Had..."
...22) are respectively the left and right Caputo-Hadamard derivatives developed in [8] and [12]....
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...The authors in [8, 12] defined the Caputo-Hadamard fractional derivatives as: • The left Caputo-Hadamard fractional of order α > 0 starting from a has the following form...
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TL;DR: In this article, the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting has been studied and several new related results are presented.
Abstract: This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.
174 citations
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
Book•
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.
7,096 citations
Book•
01 Jan 2006TL;DR: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems, which is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research.
Abstract: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.
1,609 citations
Posted Content•
TL;DR: In this paper, the Cauchy problem for the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation.
Abstract: We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order � ∈ (0,2] and skewness � (|�| ≤ min {�,2 − �}), and the first-order time derivative with a Caputo derivative of order � ∈ (0,2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < � ≤ 2, � = 1}, time-fractional diffusion {� = 2, 0 < � ≤ 2}, and neutral-fractional diffusion {0 < � = � ≤ 2}, for which the fundamental solution can be interpreted as a spatial probability density function evolving
793 citations