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Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation  [PDF]
Igor Podlubny
Physics , 2001,
Abstract: A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.
A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration  [PDF]
Nadia Benkhettou,Artur M. C. Brito da Cruz,Delfim F. M. Torres
Mathematics , 2014, DOI: 10.1016/j.sigpro.2014.05.026
Abstract: We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.
Fractional differentiation based image processing  [PDF]
Amelia Carolina Sparavigna
Computer Science , 2009,
Abstract: There are many resources useful for processing images, most of them freely available and quite friendly to use. In spite of this abundance of tools, a study of the processing methods is still worthy of efforts. Here, we want to discuss the possibilities arising from the use of fractional differential calculus. This calculus evolved in the research field of pure mathematics until 1920, when applied science started to use it. Only recently, fractional calculus was involved in image processing methods. As we shall see, the fractional calculation is able to enhance the quality of images, with interesting possibilities in edge detection and image restoration. We suggest also the fractional differentiation as a tool to reveal faint objects in astronomical images.
Fractional differentiation matrices with applications  [PDF]
Fanhai Zeng,Changpin Li
Mathematics , 2014,
Abstract: In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices are investigated numerically. The spectral collocation schemes are illustrated to solve the fractional ordinary differential equations and fractional partial differential equations. Numerical examples are also presented to illustrate the effectiveness of the derived methods, which show better performances over some existing methods.
Fractional powers of operators of Tsallis ensemble and their parameter differentiation
Rajagopal, A. K.;
Brazilian Journal of Physics , 1999, DOI: 10.1590/S0103-97331999000100006
Abstract: we develop four identities concerning parameter differentiation of fractional powers of operators appearing in the tsallis ensembles of quantum statistical mechanics of nonextensive systems. in the appropriate limit these reduce to the corresponding differentiation identities of exponential operators of the gibbs ensembles of extensive systems derived by wilcox.
Fractional powers of operators of Tsallis ensemble and their parameter differentiation  [cached]
Rajagopal A. K.
Brazilian Journal of Physics , 1999,
Abstract: We develop four identities concerning parameter differentiation of fractional powers of operators appearing in the Tsallis ensembles of quantum statistical mechanics of nonextensive systems. In the appropriate limit these reduce to the corresponding differentiation identities of exponential operators of the Gibbs ensembles of extensive systems derived by Wilcox.
Fractional order differentiation by integration with Jacobi polynomials  [PDF]
Da-Yan Liu,Olivier Gibaru,Wilfrid Perruquetti,Taous-Meriem Laleg-Kirati
Mathematics , 2012,
Abstract: The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises.
Fourier Truncation Method for Fractional Numerical Differentiation  [PDF]
Ailin Qian, Jianfeng Mao
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.27124
Abstract: We consider a ill-posed problem-fractional numerical differentiation with a new method. We propose Fourier truncation method to compute fractional numerical derivatives. A Holder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.
Fractional dynamical systems and applications in mechanics and economics  [PDF]
Mihai Boleantu,Dumitru Opris
Mathematics , 2007,
Abstract: Using the fractional integration and differentiation on R we build the fractional jet fibre bundle on a differentiable manifold and we emphasize some important geometrical objects. Euler-Lagrange fractional equations are described. Some significant examples from mechanics and economics are presented.
Fractional Directional Differentiation and Its Application for Multiscale Texture Enhancement
Chaobang Gao,Jiliu Zhou,Weihua Zhang
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/325785
Abstract: This paper derives the directional derivative expression of Taylor formula for two-variable function from Taylor formula of one-variable function. Further, it proposes a new concept, fractional directional differentiation (FDD), and corresponding theories. To achieve the numerical calculation, the paper deduces power series expression of FDD. Moreover, the paper discusses the construction of FDD mask in the four quadrants, respectively, for digital image. The differential coefficients of every direction are not the same along the eight directions in the four quadrants, which is the biggest difference by contrast to general fractional differentiation and can reflect different fractional change rates along different directions, and this benefits to enlarge the differences among the image textures. Experiments show that, for texture-rich digital images, the capability of nonlinearly enhancing comprehensive texture details by FDD is better than those by the general fractional differentiation and Butterworth filter. By quantity analysis, it shows that state-of-the-art effect of texture enhancement is obtained by FDD.
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