scholarly journals Ambiguous notation of Grünwald-Letnikov differintegral

2019 ◽  
Vol 20 (1-2) ◽  
pp. 176-179
Author(s):  
Radosław Cioć
Keyword(s):  
Lower Order ◽  
Order Derivative ◽  
Integer Order ◽  
Lower Order Derivative

The paper discussed the problem of Grünwald-Letnikov differintegral notation in which non-integer order can be incorrectly interpreted as a higher or lower order derivative. Taking the problem into consideration the author’s proposal is new notation of differintegrals.

scholarly journals Comparative study of heat and mass transfer of generalized MHD Oldroyd-B bio-nano fluid in a permeable medium with ramped conditions

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Fuzhang Wang ◽  
Sadique Rehman ◽  
Jamel Bouslimi ◽  
Hammad Khaliq ◽  
Muhammad Imran Qureshi ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Prandtl Number ◽  
Heat And Mass Transfer ◽  
Differential Operators ◽  
Fluid Model ◽  
Nano Particles ◽  
Order Derivative ◽  
Grashof Number ◽  
Integer Order ◽  
Nano Fluid

AbstractThis article aims to investigate the heat and mass transfer of MHD Oldroyd-B fluid with ramped conditions. The Oldroyd-B fluid is taken as a base fluid (Blood) with a suspension of gold nano-particles, to make the solution of non-Newtonian bio-magnetic nanofluid. The surface medium is taken porous. The well-known equation of Oldroyd-B nano-fluid of integer order derivative has been generalized to a non-integer order derivative. Three different types of definitions of fractional differential operators, like Caputo, Caputo-Fabrizio, Atangana-Baleanu (will be called later as $$C,CF,AB$$ C , C F , A B ) are used to develop the resulting fractional nano-fluid model. The solution for temperature, concentration, and velocity profiles is obtained via Laplace transform and for inverse two different numerical algorithms like Zakian’s, Stehfest’s are utilized. The solutions are also shown in tabular form. To see the physical meaning of various parameters like thermal Grashof number, Radiation factor, mass Grashof number, Schmidt number, Prandtl number etc. are explained graphically and theoretically. The velocity and temperature of nanofluid decrease with increasing the value of gold nanoparticles, while increase with increasing the value of both thermal Grashof number and mass Grashof number. The Prandtl number shows opposite behavior for both temperature and velocity field. It will decelerate both the profile. Also, a comparative analysis is also presented between ours and the existing findings.

Analysis of a Non-integer Order Model for the Coinfection of HIV and HSV-2

2020 ◽  
Vol 21 (3-4) ◽  
pp. 291-302
Author(s):  
Carla M. A. Pinto ◽  
Ana R. M. Carvalho
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Point Of View ◽  
Order Derivative ◽  
Order Model ◽  
Integer Order ◽  
Disease Free

AbstractWe propose a non-integer order model for the dynamics of the coinfection of HIV and HSV-2. We calculate the reproduction number of the model and study the local stability of the disease-free equilibrium. Simulations of the model for the variation of epidemiologically relevant parameters and the order of the non-integer order derivative, α, reveal interesting dynamics. These results are discussed from an epidemiologically point of view.

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

2020 ◽  
Vol 234 (24) ◽  
pp. 4801-4812 ◽  
Author(s):  
Rajendra K Praharaj ◽  
Nabanita Datta
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Load ◽  
Order Derivative ◽  
Viscoelastic Foundation ◽  
Bernoulli Beam ◽  
Integer Order ◽  
Foundation Model ◽  
Euler Bernoulli Beam

The dynamic behaviour of an Euler–Bernoulli beam resting on the fractionally damped viscoelastic foundation subjected to a moving point load is investigated. The fractional-order derivative-based Kelvin–Voigt model describes the rheological properties of the viscoelastic foundation. The Riemann–Liouville fractional derivative model is applied for a fractional derivative order. The modal superposition method and Triangular strip matrix approach are applied to solve the fractional differential equation of motion. The dependence of the modal convergence on the system parameters is studied. The influences of (a) the fractional order of derivative, (b) the speed of the moving point load and (c) the foundation parameters on the dynamic response of the system are studied and conclusions are drawn. The damping of the beam-foundation system increases with increasing the order of derivative, leading to a decrease in the dynamic amplification factor. The results are compared with those using the classical integer-order derivative-based foundation model. The classical foundation model over-predicts the damping and under-predicts the dynamic deflections and stresses. The results of the classical (integer-order) foundation model are verified with literature.

scholarly journals Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel

2016 ◽  
Vol 55 (3) ◽  
pp. 2789-2796 ◽  
Author(s):  
A.A. Zafar ◽  
C. Fetecau
Keyword(s):  
Viscous Fluid ◽  
Infinite Plate ◽  
Singular Kernel ◽  
Order Derivative ◽  
Integer Order

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

Edge enhancement of magnetic data using fractional-order-derivative filters

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. J7-J17 ◽  
Author(s):  
Muzaffer Özgü Arısoy ◽  
Ünal Dikmen
Keyword(s):  
Fractional Order ◽  
Order Derivative ◽  
Magnetic Data ◽  
Frequency Noise ◽  
Edge Enhancement ◽  
Traditional Use ◽  
Data Set ◽  
Fractional Order Derivative ◽  
Detection Techniques ◽  
Integer Order

Edge enhancement and detection techniques are fundamental operations in magnetic data interpretation. Many techniques for edge enhancement have been developed, some based on profile data and others designed for grid-based data sets. Methods that are traditionally applied to magnetic data, such as total horizontal derivative (THD) and analytic signal (AS), require the computation of integer-order horizontal and vertical derivatives of the magnetic data. However, if the data set contains features with a large variation in amplitude, then the features with small amplitudes may be difficult to outline. In addition, because most edge enhancement and detection filters are derivative-based filters, they also amplify high-frequency noise content in the data. As a result, the accuracy of derivative-based filters is restricted to data of high quality. We suggested the modification of the THD and AS filters by combining the amplitude spectra of fractional-order-derivative filters with ad hoc phase spectra, particularly designed for edge detection in magnetic data. We revealed the capability of the proposed algorithm on synthetic magnetic data and on aeromagnetic data from Turkey. Compared with the traditional use of THD and AS (with integer-order derivatives), we developed the method based on fractional-order derivatives that produced more effective results in terms of suppressing noise and delineating the edges of deep sources.

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