scholarly journals Modeling and Simulation of Equivalent Circuits in Description of Biological Systems - A Fractional Calculus Approach

2019 ◽  
Vol 3 (1) ◽  
pp. 2-11 ◽  
Author(s):  
F. Gómez ◽  
J. Bernal ◽  
J. Rosales ◽  
T. Cordova
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Transfer Functions ◽  
Electrical Circuit ◽  
Equivalent Electrical Circuit ◽  
Electrical Parameters ◽  
Special Cases ◽  
Fractional Differential ◽  
Cole Model

Abstract Using the fractional calculus approach, we present the Laplace analysis of an equivalent electrical circuit for a multilayered system, which includes distributed elements of the Cole model type. The Bode graphs are obtained from the numerical simulation of the corresponding transfer functions using arbitrary electrical parameters in order to illustrate the methodology. A numerical Laplace transform is used with respect to the simulation of the fractional differential equations. From the results shown in the analysis, we obtain the formula for the equivalent electrical circuit of a simple spectrum, such as that generated by a real sample of blood tissue, and the corresponding Nyquist diagrams. In addition to maintaining consistency in adjusted electrical parameters, the advantage of using fractional differential equations in the study of the impedance spectra is made clear in the analysis used to determine a compact formula for the equivalent electrical circuit, which includes the Cole model and a simple RC model as special cases.

scholarly journals Generalized proportional fractional integral Hermite–Hadamard’s inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tariq A. Aljaaidi ◽  
Deepak B. Pachpatte ◽  
Thabet Abdeljawad ◽  
Mohammed S. Abdo ◽  
Mohammed A. Almalahi ◽  
...  
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Approximation Theory ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Fractional Operators ◽  
Uniqueness Of Solutions ◽  
Special Cases ◽  
Fractional Differential

AbstractThe theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

scholarly journals Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 701 ◽  
Author(s):  
Suphawat Asawasamrit ◽  
Sotiris Ntouyas ◽  
Jessada Tariboon ◽  
Woraphak Nithiarayaphaks
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Hadamard Fractional Differential Equations ◽  
Special Cases ◽  
Separated Boundary Conditions ◽  
Fractional Differential

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.

scholarly journals Boundary value problems for Hilfer fractional differential equations with Katugampola fractional integral and anti-periodic conditions

Mathematica ◽  
2021 ◽  
Vol 63 (86) (2) ◽  
pp. 208-221
Author(s):  
Abdelatif Boutiara ◽  
◽  
Maamar Benbachir ◽  
Kaddour Guerbati ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Integral Boundary Conditions ◽  
General Nature ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
Special Cases ◽  
Fractional Differential

The purpose of this paper is to investigate the existence and uniqueness of solutions for a new class of nonlinear fractional differential equations involving Hilfer fractional operator with fractional integral boundary conditions. Our analysis relies on classical fixed point theorems and the Boyd-Wong nonlinear contraction. At the end, an illustrative example is presented. The boundary conditions introduced in this work are of quite general nature and can be reduce to many special cases by fixing the parameters involved in the conditions.

scholarly journals Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

2016 ◽  
Vol 14 (1) ◽  
pp. 723-735 ◽  
Author(s):  
Mohammed H. Aqlan ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Juan J. Nieto
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Integral Boundary Conditions ◽  
Existence Theory ◽  
Integral Boundary ◽  
Special Cases ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractWe develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.

scholarly journals On a Fractional Operator Combining Proportional and Classical Differintegrals

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 360 ◽  
Author(s):  
Dumitru Baleanu ◽  
Arran Fernandez ◽  
Ali Akgül
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Inverse Operator ◽  
Fractional Operator ◽  
Special Cases ◽  
Non Local ◽  
Fractional Differential

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

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