Conformable fractional derivative and its application to partial fractional derivatives

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

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A solution method for integro-differential equations of conformable fractional derivative

Thermal Science ◽

10.2298/tsci170624266b ◽

2018 ◽

Vol 22 (Suppl. 1) ◽

pp. 7-14 ◽

Cited By ~ 2

Author(s):

Mustafa Bayram ◽

Veysel Hatipoglu ◽

Sertan Alkan ◽

Sebahat Das

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Fractional Derivative ◽

Exact Solutions ◽

Fractional Derivatives ◽

Solution Method ◽

Approximate Solutions ◽

Conformable Fractional Derivative ◽

Conformable Derivative ◽

Numerical Examples

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

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Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

Thermal Science ◽

10.2298/tsci200520035f ◽

2021 ◽

pp. 35-35

Author(s):

Jing-Li Fu ◽

Lijun Zhang ◽

Chaudry Khalique ◽

Ma-Li Guo

Keyword(s):

Fractional Derivative ◽

Fractional Derivatives ◽

Lagrangian Systems ◽

Conserved Quantities ◽

Conformable Fractional Derivative ◽

Noether Symmetries ◽

Hamilton’S Principle ◽

Hamilton's Principle ◽

Motion Equations ◽

Definition Of

In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton's principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamilton's principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem and Hojman's conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.

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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4037930 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 8

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.

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Fractional-Order Colour Image Processing

Mathematics ◽

10.3390/math9050457 ◽

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.

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

Electronics ◽

10.3390/electronics10040475 ◽

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.

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Exponential Stability of Hopfield Neural Networks with Conformable Fractional Derivative

Neurocomputing ◽

10.1016/j.neucom.2021.05.076 ◽

2021 ◽

Author(s):

Aysen Kütahyalıoglu ◽

Fatma Karakoç

Keyword(s):

Neural Networks ◽

Fractional Derivative ◽

Exponential Stability ◽

Hopfield Neural Networks ◽

Conformable Fractional Derivative

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

Open Physics ◽

10.2478/s11534-011-0040-5 ◽

2011 ◽

Vol 9 (5) ◽

Cited By ~ 4

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.

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

Mathematics ◽

10.3390/math9090979 ◽

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.

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Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽

10.1515/phys-2019-0088 ◽

2019 ◽

Vol 17 (1) ◽

pp. 850-856 ◽

Cited By ~ 2

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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