Definition Of Fractional Derivative Research Articles

Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method

In this paper, an analytical and numerical algorithm for solving displacement response of fractional oscillator system with cosine excitation is established. The analytical method is to solve steady‐state response and transient response solutions of the system by means of average method and then the total displacement response solution is the sum of steady‐state solution and transient solution. The numerical method is to use the Grünwald–Letnikov definition of fractional derivatives to discretize the fractional differential term in the system and then to reduce the order of the original system. Then, the approximate response solution of the system under the general periodic excitation is obtained by using the Fourier series expansion method and the superposition principle of linear system. Finally, the effectiveness and feasibility of the analytical technique are verified by numerical simulation, and the influence of fractional order, linear damping coefficient and fractional derivative coefficient on the amplitude of the steady‐state response and the total displacement response of the system is analyzed.

An efficient technique to analyze the fractional model of vector-borne diseases

In the present work, we find and analyze the approximated analytical solution for the vector-borne diseases model of fractional order with the help of -homotopy analysis transform method (-HATM). Many novel definitions of fractional derivatives have been suggested and utilized in recent years to build mathematical models for a wide range of complex problems with nonlocal effects, memory, or history. The primary goal of this work is to create and assess a Caputo–Fabrizio fractional derivative model for Vector-borne diseases. In this investigation, we looked at a system of six equations that explain how vector-borne diseases evolve in a population and how they affect community public health. With the influence of the fixed-point theorem, we establish the existence and uniqueness of the models system of solutions. Conditions for the presence of the equilibrium point and its local asymptotic stability are derived. We discover novel approximate solutions that swiftly converge. Furthermore, the future technique includes auxiliary parameters that are both trustworthy and practical for managing the convergence of the solution found. The current study reveals that the investigated model is notably dependent on the time chronology and also the time instant, which can be effectively studied with the help of the arbitrary order calculus idea.

Electrical Circuits Described by General Fractional Conformable Derivative

The general fractional conformable derivative (GCD) and its attributes have been described by researchers in the recent times. Compared with other fractional derivative definitions, this derivative presents a generalization of the conformable derivative and follows the same derivation formulae. For electrical circuits, such as RLC, RC, and LC, we obtain a new class of fractional-order differential equations using this novel derivative, The use of GCD to depict electrical circuits has been shown to be more adaptable and lucrative than the usual conformable derivative.

Impact of Al2O3 in Electrically Conducting Mineral Oil-Based Maxwell Nanofluid: Application to the Petroleum Industry

Alumina nanoparticles (Al2O3) are one of the essential metal oxides and have a wide range of applications and unique physio-chemical features. Most notably, alumina has been shown to have thermal properties such as high thermal conductivity and a convective heat transfer coefficient. Therefore, this study is conducted to integrate the adsorption of Al2O3 in mineral oil-based Maxwell fluid. The ambitious goal of this study is to intensify the mechanical and thermal properties of a Maxwell fluid under heat flux boundary conditions. The novelty of the research is increased by introducing fractional derivatives to the Maxwell model. There are various distinct types of fractional derivative definitions, with the Caputo fractional derivative being one of the most predominantly applied. Therefore, the fractoinal-order derivatives are evaluated using the fractional Caputo derivative, and the integer-order derivatives are evaluated using the Crank–Nicolson method. The obtained results are graphically displayed to demonstrate how all governing parameters, such as nanoparticle volume fraction, relaxation time, fractional derivative, magnetic field, thermal radiation, and viscous dissipation, have a significant impact on fluid flow and temperature distribution.

Applications of fractional calculus in computer vision: A survey

Fractional calculus is an abstract idea exploring interpretations of differentiation having non-integer order. For a very long time, it was considered as a topic of mere theoretical interest. However, the introduction of several useful definitions of fractional derivatives has extended its domain to applications. Supported by computational power and algorithmic representations, fractional calculus has emerged as a multifarious domain. It has been found that the fractional derivatives are capable of incorporating memory into the system and thus suitable to improve the performance of locality-aware tasks such as image processing and computer vision in general. This article presents an extensive survey of fractional-order derivative-based techniques that are used in computer vision. It briefly introduces the basics and presents applications of the fractional calculus in six different domains viz. edge detection, optical flow, image segmentation, image de-noising, image recognition, and object detection. The fractional derivatives ensure noise resilience and can preserve both high and low-frequency components of an image. The relative similarity of neighboring pixels can get affected by an error, noise, or non–homogeneous illumination in an image. In that case, the fractional differentiation can model special similarities and help compensate for the issue suitably. The fractional derivatives can be evaluated for discontinuous functions, which help estimate discontinuous optical flow. The order of the differentiation also provides an additional degree of freedom in the optimization process. This study shows the successful implementations of fractional calculus in computer vision and contributes to bringing out challenges and future scopes.

An Application of Homotopy Perturbation Method to Fractional-Order Thin Film Flow of the Johnson–Segalman Fluid Model

Thin film flow is an important theme in fluid mechanics and has many industrial applications. These flows can be observed in oil refinement process, laser cutting, and nuclear reactors. In this theoretical study, we explore thin film flow of non-Newtonian Johnson–Segalman fluid on a vertical belt in fractional space in lifting and drainage scenarios. Modelled fractional-order boundary value problems are solved numerically using the homotopy perturbation method along with Caputo definition of fractional derivative. In this study, instantaneous and average velocities and volumetric flux are computed in lifting and drainage cases. Validity and convergence of homotopy-based solutions are confirmed by finding residual errors in each case. Moreover, the consequences of different fractional and fluid parameters are graphically studied on the velocity profile. Analysis shows that fractional parameters have opposite effects of the fluid velocity.

Thermal applications of copper oxide, silver, and titanium dioxide nanoparticles via fractional derivative approach

The recent progress in nanotechnology developed more stable nanofluid models with ultra-high thermal performances. The immersion of hybrid nanoparticles in the base fluid presents novel applications in the thermal transport processes, solar energy, engineering devices, and heating systems. This continuation identified the thermal outcomes of the hybrid nanofluid model with the water base fluid. The thermal investigation is encountered by utilizing the copper oxide (CuO), silver (Ag), and titanium dioxide (TiO2) nanoparticles. The thermal selection of these three types of nanoparticles is associated with impressive characteristics and stability. To attain the more progressive thermal aspects, thermal radiation consequences are also focused. The inclined infinite configuration is assumed where these nanoparticles and base fluid are specified. The comparative results for the hybrid nanofluid and modified hybrid models are pronounced. The fractional model of dimensionless partial differential leading equations is formulated by following the recent definitions of fractional derivatives, namely, Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. The semi-analytical simulations are performed via the Laplace technique for nanoparticles temperature and velocity inspection. The significances of the fractional parameter and volume fraction are observed for the velocity profile.

Thermoelastic diffusion responses of sandwich structures associated with new definitions of fractional derivative

The last five years have witnessed great developments of tempered fractional diffusion equation, both theoretically and numerically. In this work, new insights on fractional order thermoelastic diffusion are provided in two aspects: first, a novel form of fractional order model is proposed directly from Fourier’s law and Fick’s law; second, tempered fractional derivative is adopted, which shares the property that: it may transit from power-law behavior to exponential behavior in large time scale. The new model is applied to study the transient responses of a sandwich structure with ideal interfacial conditions subjected to symmetrical thermal and chemical shock loadings. In the numerical part, Laplace transform method is adopted, and the effects of different fractional order definitions, material characteristic parameters and fractional derivative parameters on the thermoelastic diffusion fields are analyzed and presented graphically. The results show that different fractional order definitions have different memory effects on the transient responses, and the effects of fractional order definitions on the responses will vary accordingly with the parameter values. This may provide insight to optimize the thermodiffusion management by selecting different materials and fractional derivative forms.

Analysis of HIV/AIDS model with Mittag-Leffler kernel

<abstract><p>Recently different definitions of fractional derivatives are proposed for the development of real-world systems and mathematical models. In this paper, our main concern is to develop and analyze the effective numerical method for fractional order HIV/ AIDS model which is advanced approach for such biological models. With the help of an effective techniques and Sumudu transform, some new results are developed. Fractional order HIV/AIDS model is analyzed. Analysis for proposed model is new which will be helpful to understand the outbreak of HIV/AIDS in a community and will be helpful for future analysis to overcome the effect of HIV/AIDS. Novel numerical procedures are used for graphical results and their discussion.</p></abstract>

Caputo definition for finding fractional moments of power law distribution functions

In this paper, we propose a modern technique to derive fractional moment by using Caputo definition of fractional derivative. Such technique represents a modification of fractional moments on a certain type of function which is known by the power law distribution functions. The results have been obtained show that the obtained fractional moments within such functions have closed form.

A variety of fractional soliton solutions for three important coupled models arising in mathematical physics

This paper deals with the optical solitons of fractional coupled Boussinesq, Burgers-type and mKdV equations by the hypothesis of traveling wave and [Formula: see text]-expansion scheme. These equations are important in different fields such as propagation of long water waves, fluid dynamics, and shallow water wave propagation. In comparison to other analytical procedures, the analytical methodology [Formula: see text] is an incredibly beneficial approach. This technique can also be used with other nonlinear fractional models. The suggested method generates three distinct solutions such as trigonometric, hyperbolic, and rational. Moreover, graphical representation has been used to portray the physical significance of the constructed solutions. Finally, a comprehensive study is made by using a definition of Beta fractional derivative and obtained solutions are represented graphically to understand considered models.

A New Conformable Fractional Derivative and Applications

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 .

A fractional taylor series-Based least mean square algorithm, and its application to power signal estimation

The definition of fractional derivative by Caputo and Riemann inspired the researchers to develop new adaptive algorithms having better convergence properties in comparison with integer-gradient based adaptive algorithms. As reported in many studies, the existing fractional gradient-based adaptive algorithms lack justification for using fractional derivative in addition to integer-gradient, may become inconsistent in the event of negative weights, and may yield more or less same performance as compared to the conventional integer-derivative-based algorithms by appropriate selection of step-size parameter. Accordingly, this paper presents a novel fractional adaptive algorithm based on Fractional Taylor Series. Unlike (most of) the existing fractional-derivative-based algorithms, the proposed algorithm only involves fractional-derivative and ensures convergence of mean square error (MSE) provided the step-size is chosen appropriately. Simulation results are presented in order to depict a scenario where exploitation of fractional-derivative in the weight-update equation yields better convergence as compared to LMS algorithm in the context of power signal parameters estimation.

B-Spline Collocation Discretizations of Caputo and Riemann-Liouville Derivatives: A Matrix Comparison

Two of the most famous definitions of fractional derivatives are the Riemann-Liouville and the Caputo ones. In principle, these formulations are not equivalent and ask for different levels of regularity of the considered function. By focusing on a B-spline collocation discretization of both kind of derivatives, we show that when the fractional order α ranges in (1, 2) their difference in terms of matrices corresponds to a rank-1 correction whose spectral norm increases with the mesh-size n and is (\(o(\sqrt n )\)). On one hand, this implies that the spectral distribution for the B-spline collocation matrices corresponding to the Riemann-Liouville and Caputo derivatives coincide; on the other hand, the presence of the rank-1 correction makes the Caputo matrices worse conditioned for α tending to 1 due to a larger maximum singular value. Some linear algebra consequences of all this knowledge are discussed, and a selection of numerical experiments that validate our findings is provided.

An Analytical Study for Caputo Fractional Derivative on Unsteady Casson Fluid with Thermal Radiation Effect

Studies on Casson fluid are essential in the development of the manufacturing and engineering fields since it is widely used there. Meanwhile, fractional derivative has been known to be a constructive paradox that can be beneficial in the future. In this study, the development fractional derivative on Casson fluid flow is investigated. A fractional Casson fluid model with effect of thermal radiation is derived together with momentum and energy equations. The Caputo definition of fractional derivative is used in the mathematical formulation. Casson fluid with constant wall temperature over an oscillating plate in the presence of thermal radiation is considered. Solutions were obtained by using Laplace transform and are presented in the form of Wright function. Graphical analysis on velocity and temperature profiles was conducted with variations in parametric values such as fractional parameter, Grashof number, Prandtl number and radiation parameter. Numerical computations were carried out to investigate behaviours of skin friction and Nusselt number. It is found that when the fractional parameter is increased, the velocity and temperature profiles will also increase. Existence of fractional parameter in both velocity and temperature profiles shows the transitional phenomenon of both profiles from an unsteady state to steady state, providing a new perspective on Casson fluid flow. An increment in both profiles is also observed when the thermal radiation parameter is increased. The present results are also validated with published results, and it is found that they are in agreement with each other.

A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case

In this paper, the existence results of the solution and the finite-time stability (FTS) are focused for fractional fuzzy differential equations (FFDEs) involving non-instantaneous impulsive effects and perturbance parameters. In view of the definitions of short-memory fractional derivatives and a short-memory fractional modeling approach, the existence of a unique solution is proved through utilizing the fixed point theorem of a weakly contractive mapping on the partially ordered fuzzy space, and the sufficient condition of FTS is obtained by using the generalized fractional Gronwall inequality. The fuzzy fractional derivatives used in this study include the concept of derivatives based on the generalized Hukuhara difference (gH-difference) and the granular difference (gr-difference). It is demonstrated that the obtained results based on the gr-difference can overcome restrictions associated with the previous approaches defined via the gH-difference concept. Finally, some numerical examples are provided to verify our main results.

New Solitary and Periodic Wave Solutions of (n + 1)-Dimensional Fractional Order Equations Modeling Fluid Dynamics

In this study, first, fractional derivative definitions in the literature are examined and their disadvantages are explained in detail. Then, it seems appropriate to apply the (G′G)-expansion method under Atangana’s definition of β-conformable fractional derivative to obtain the exact solutions of the space–time fractional differential equations, which have attracted the attention of many researchers recently. The method is applied to different versions of (n+1)-dimensional Kadomtsev–Petviashvili equations and new exact solutions of these equations depending on the β parameter are acquired. If the parameter values in the new solutions obtained are selected appropriately, 2D and 3D graphs are plotted. Thus, the decay and symmetry properties of solitary wave solutions in a nonlocal shallow water wave model are investigated. It is also shown that all such solitary wave solutions are symmetrical on both sides of the apex. In addition, a close relationship is established between symmetric and propagated wave solutions.

Numerical Method to Solve a Hybrid Fuzzy Conformable Fractional Differential Equations

This research introduces a new definition of fuzzy fractional derivative, fuzzy conformable fractional derivative, which is defined based on generalized Hukuhara differentiability. Namely, we investigate the Hybrid fuzzy fractional differential equation with the fuzzy conformable fractional generalized Hukuhara derivative. We establish that the Hybrid fuzzy fractional differential equation admits two fuzzy triangular solutions and prove that these fuzzy solutions are obtained together with a characterization of these solutions by two systems of fractional differential equations. We propose an adaptable numerical scheme for the approximation of the fuzzy triangular solutions. Numerical results reveal that the numerical method is convenient for solving the Hybrid fuzzy conformable fractional differential equation.

A Generalized Definition of the Fractional Derivative with Applications

A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function expanded by a Taylor series, we show that D α D β f t = D α + β f t ; 0 < α ≤ 1 ; 0 < β ≤ 1 . GFD is applied for some functions to investigate that the GFD coincides with the results from Caputo and Riemann–Liouville fractional derivatives. The solutions of the Riccati fractional differential equation are obtained via the GFD. A comparison with the Bernstein polynomial method BPM , enhanced homotopy perturbation method EHPM , and conformable derivative CD is also discussed. Our results show that the proposed definition gives a much better accuracy than the well-known definition of the conformable derivative. Therefore, GFD has advantages in comparison with other related definitions. This work provides a new path for a simple tool for obtaining analytical solutions of many problems in the context of fractional calculus.

Natural convection flow of radiative maxwell fluid with Newtonian heating and slip effects: Fractional derivatives simulations

The unsteady natural convection flow for the maxwell fluid due to infinite plate is inspected with help off ractional derivatives. The effects of slip conditions, Newtonian heating effect, MHD, and radiation are also considered. The recent definitions of fractional derivatives i.e. Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives are used to construct the fractional modal of leading partial differential equations. The semi-analytical solution of the governing equations is attained by utilizing the Laplace transformation and some numerical techniques i.e. Stehfest and Tzou's algorithms. Moreover, to analyze the physical significance of the under consideration problem, the graphical analysis is explored. The results concluded that the temperature and velocity profile obtained via CF-fractional derivative shows a declining trend when compared with AB-fractional derivative. Furthermore, the velocity and temperature fields were also decayed by varying the value of fractional parameters in both AB and CF cases.