Fractional calculus via Laplace transform and its application in relaxation processes

@article{Oliveira2019FractionalCV,
  title={Fractional calculus via Laplace transform and its application in relaxation processes},
  author={E. Capelas De Oliveira and Stef{\^a}nia Jarosz and Jayme Vaz},
  journal={Commun. Nonlinear Sci. Numer. Simul.},
  year={2019},
  volume={69},
  pages={58-72}
}
A new Laplace-type fractional derivative
In this paper, we present a new derivative via the Laplace transform. The Laplace transform leads to a natural form of the fractional derivative which is equivalent to a Riemann-Liouville derivative
Anomalous relaxation in dielectrics with Hilfer fractional derivative
We introduce a new relaxation function depending on an arbitrary parameter as solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole,
3 J an 2 02 0 A FRACTIONAL DERIVATIVE WITH A NEW SMOOTH KERNEL
Fractional calculus has developed remarkably in recent years and used in physics, engineering, economics etc [1, 2, 3]. Classical results about the Riemann-Liouville and Caputo derivatives as well as
On the mistake in defining fractional derivative using a non-singular kernel
Definitions of fractional derivative of order $\alpha$ ($0 < \alpha \leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in
Analytical Solutions for Nonlinear Fractional Physical Problems Via Natural Homotopy Perturbation Method
  • A. Arafa
  • Mathematics
    International Journal of Applied and Computational Mathematics
  • 2021
The main objective of this paper is to describe new analytical solutions for the nonlinear fractional Fisher equation, the nonlinear fractional Boussinesq-like equation and the nonlinear fractional
The Volterra type equations related to the non-Debye relaxation
The solutions of some certain non-homogeneous fractional integral equations
In this paper, we propose the solutions of non-homogeneous fractional integral equations of the form I 2σ 0+ y(t) +a·I σ 0+ y(t) +b·y(t)=t n , and I 2σ 0+ y(t) +a·I σ 0+ y(t) +b·y(t)=t n e t , where
Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel
Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional
Mittag–Leffler Memory Kernel in Lévy Flights
In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of
Analytic approaches of the anomalous diffusion: A review
...
...

References

SHOWING 1-10 OF 85 REFERENCES
Fractional Calculus: Integral and Differential Equations of Fractional Order
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the
Fractional calculus and its applications
  • Changpin Li, Y. Chen, J. Kurths
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2013
TLDR
This Theme Issue, including one review article and 12 research papers, can be regarded as a continuation of the first special issue of European Physical Journal Special Topics in 2011, and the second specialissue of International Journal of Bifurcation and Chaos in 2012.
Fox function representation of non-debye relaxation processes
Applying the Liouville-Riemann fractional calculus, we derive and solve a fractional operator relaxation equation. We demonstrate how the exponentΒ of the asymptotic power law decay ∼t−β relates to
Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels
TLDR
For these new mode ls, the coherence with the thermodynamic laws is proved and the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism is revised by these new fractional operators.
Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application
Fractional Calculus and Fractional Processes with Applications to Financial Economics presents the theory and application of fractional calculus and fractional processes to financial data. Fractional
Fractional kinetic equations: solutions and applications.
TLDR
Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed, presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process.
Applications Of Fractional Calculus In Physics
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences,
On initial conditions, generalized functions and the Laplace transform
This note exposes the mathematical setting of initial value problems for causal time-invariant linear systems, given by ordinary differential equations within the framework of generalized functions.
Fourier and Laplace Transforms
This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of
Mittag-Leffler Functions, Related Topics and Applications
As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions
...
...