Dynamics of a Fractional Derivative Type of a Viscoelastic Rod With Random Excitation

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Teodor Atanacković ◽  
Marko Nedeljkov ◽  
Stevan Pilipović ◽  
Danijela Rajter-Ćiri
Keyword(s):  
Fractional Derivative ◽  
Weak Solutions ◽  
Constitutive Equations ◽  
Fractional Derivatives ◽  
Random Excitation ◽  
Bounded Noise ◽  
White Noise Process ◽  
Derivative Type ◽  
Axial Vibrations ◽  
Viscoelastic Rod

AbstractThe axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations as well as their stochastic moments are determined.

scholarly journals A generalization of truncated M-fractional derivative and applications to fractional differential equations

2020 ◽  
Vol 5 (1) ◽  
pp. 171-188 ◽  
Author(s):  
Esin İlhan ◽  
İ. Onur Kıymaz
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integer Order ◽  
Derivative Type ◽  
Fractional Differential

AbstractIn this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

scholarly journals On a fractional derivative type of a viscoelastic body

2002 ◽  
pp. 27-38 ◽  
Author(s):  
Teodor Atanackovic ◽  
Branislava Novakovic
Keyword(s):  
Stress State ◽  
Fractional Derivative ◽  
Constitutive Equations ◽  
Viscoelastic Body ◽  
Special Cases ◽  
Derivative Type

We study a viscoelastic body, in a linear stress state with fractional derivative type of dissipation. The model was formulated in [1]. Here we derive restrictions on the model that follow from Clausius-Duhem inequality. Several known constitutive equations are derived as special cases of our model. Two examples are discussed. .

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

scholarly journals Constant curvature coefficients and exact solutions in fractional gravity and geometric mechanics

Open Physics ◽  
2011 ◽  
Vol 9 (5) ◽  
Author(s):  
Dumitru Baleanu ◽  
Sergiu Vacaru
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Geometric Mechanics ◽  
Fractional Dynamics ◽  
Constant Matrix ◽  
Curve Flows ◽  
Physical Interactions ◽  
Nonholonomic Manifolds ◽  
Lagrange Mechanics ◽  
Fractional Gravity

AbstractWe present a study of fractional configurations in gravity theories and Lagrange mechanics. The approach is based on a Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists of a proof that, for corresponding classes of nonholonomic distributions, a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

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