A new definition of fractional derivative

Zheng, Zhibao; Zhao, Wei; Dai, Hongzhe

doi:10.1016/j.ijnonlinmec.2018.10.001

Cited by 40 publications

(21 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, the fractional calculus became an important tool for biology, chemistry, physics and many engineering domains. A good presentation of the theory of fractional operators, together with their properties and applications can be found in Hristov,13,14 Povstenko, 15,16 Zheng et al, 17 Baleanu et al 18,19 and Hilfer. 20 The purpose of this article is to study two-layer unsteady one-dimensional flows of incompressible and immiscible generalized second grade fluids in a rectangular channel.…”

Section: Introductionmentioning

confidence: 99%

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

Luo

Alarifi

Vieru

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t. K E Y W O R D S fractional calculus, immiscible fluids, memory effects, two-layer flow

show abstract

Section: Introductionmentioning

confidence: 99%

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

Luo

Alarifi

Vieru

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The maximum value associated with this 'WT-map' is 87.5% which occurs at (W, T) coordinates (40,10). From Figure 8 it can be seen that the highest combined accuracies (>70%) are obtained for approximate values of W ∈ [30,50] and T ∈ [10,20]. However, it should be noted that WT-maps of this kind are data dependent and will vary with the type of financial time series that is processed and on the non-stationary characteristics that occur over the length of the data series that is chosen (i.e., the input parameter L in function Backtester).…”

Section: Example Resultsmentioning

confidence: 99%

“…Note however, that the "process" of generalising the Fourier transform used above for defining fractional differentials and integrals is just one such generalisation that can be applied. Thus, the operators defined by Equations (5) and (6), for example, are not unique and there are many definitions and generalisations of a fractional derivative that have been developed [19] and continue to be so [20].…”

Section: Fractional Integrals and Differentialsmentioning

confidence: 99%

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

et al. 2019

View full text Add to dashboard Cite

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index γ ∈ [ 0 , 2 ] and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

show abstract

“…Fractional calculus and fractal calculus have become the hotspots in the fields of mathematics, physics, and engineering. In order to further study the nonlinear propagation of ion-acoustic waves in ultrarelativistic plasmas plasma, in this section, we use the semiinverse method and fractional variation principle to derive the space-time fractional order equations from equations (10) and (20).…”

Section: Derivation Of Time-space Fractional Cylindrical Equationsmentioning

confidence: 99%

“…Most of them first introduce fractional order integrals and define fractional derivatives on this basis. However, after a series of improvements, the concept of fractional derivatives [20][21][22] is more suitable for the modeling of practical problems than the integer derivatives. Fractional differential equations [23][24][25] are also beginning to be used to solve problems in the fields of biological systems, thermal systems, and mechanical systems, and their practicality is getting stronger.…”

Section: Introductionmentioning

confidence: 99%

Time-Space Fractional Model for Complex Cylindrical Ion-Acoustic Waves in Ultrarelativistic Plasmas

Liu

Chen

2020

Complexity

View full text Add to dashboard Cite

In this paper, the fractional order models are used to study the propagation of ion-acoustic waves in ultrarelativistic plasmas in nonplanar geometry (cylindrical). Firstly, according to the control equations, (2 + 1)-dimensional (2D) cylindrical Kadomtsev–Petviashvili (CKP) equation and 2D cylindrical-modified Kadomtsev–Petviashvili (CMKP) equation are derived by using multiscale analysis and reduced perturbation methods. Secondly, using the semi-inverse method and the fractional variation principle, the abovementioned equations are derived the time-space fractional equations (TSF-CKP and TSF-CMKP). Furthermore, based on the fractional order transformation, the 1-decay mode solution of the TSF-CKP equation is obtained by using the simplified homogeneous balance method, and using the generalized hyperbolic-function method, the exact analytic solution of TSF-CMKP equation is obtained. Finally, the effects of the phase speed λ, electron number density (through β3) and the fractional order α,β,ω on the propagation of ion-acoustic waves in ultrarelativistic plasmas are analyzed.

show abstract

A new definition of fractional derivative

Cited by 40 publications

References 33 publications

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

Two‐layer flows of generalized immiscible second grade fluids in a rectangular channel

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

Time-Space Fractional Model for Complex Cylindrical Ion-Acoustic Waves in Ultrarelativistic Plasmas

Contact Info

Product

Resources

About