Application Of Fractional Derivatives Research Articles

Dynamic Analysis and Optimal Control of a Fractional Order Fishery Model with Refuge and Protected Area

In this paper, a mathematical analysis of fractional order fishery model with stage structure for predator is carried out under the background of prey refuge and protected area. First, it is demonstrated that the solution exists and is unique. The paper aims to analyze predator-prey dynamics in a fishery model through the application of fractional derivatives. It is worth emphasizing that we explicitly examine how fractional derivatives affect the dynamics of the model. The existence of each equilibrium point and the stability of the system at the equilibrium point are proved. The theoretical results are proved by numerical simulation. Alternatively, allocate harvesting efforts within an improved model aimed at maximizing economic benefits and ecologically sustainable development. The ideal solution is obtained by applying Pontryagin’s optimal control principle. A large number of numerical simulations show that the optimal control scheme can realize the sustainable development of the ecosystem.

Application of fractional derivatives in image quality assessment indices

Objective image quality assessment involves the use of mathematical models to quantitatively describe image quality. FR-IQA (Full-Reference Image Quality Assessment) methods using reference images are also often used to evaluate image processing and computer vision algorithms. Quality indices often use gradient operators to express relevant visual information, such as edges. Fractional calculus has been applied in the last two decades in various fields such as signal processing, image processing, and pattern recognition. Fractional derivatives are generalizations of integer-order derivatives and can be computed using various operators such as the Riemann-Liouville, Caputo-Fabrizio, and Grünwald-Letnikov operators. In this paper, we propose a modification of the FSIMc image quality index by including fractional derivatives to extract and enhance edges. A study of the usefulness of fractional derivative in the FSIMc model was conducted by assessing Pearson, Spearman and Kendall correlations with MOS scores for images from the TID2013 and KADID-10k databases. Comparison of FD_FSIMc with the classic FSIMc shows an increase of several percent in the correlation coefficients for the modified index. The results obtained are superior to those other known approaches to FR-IQA that use fractional derivatives. The results encourage the use of fractional calculus.

Application of fractional derivative on nanofluid with magnetic field

The study investigates the combined impact of heat and mass transfer on a vertical plate, considering factors such as an applied magnetic field, thermal radiation, heat absorption, and a first-order chemical reaction. The fluid under examination is assumed to be an electrically conducting Cu-nanofluid based on water. Thermophysical properties are derived from Table 1 for both the base fluid (water) and nanoparticles (Cu), with the nanoparticle volume fraction chosen within the range of 0.01 to 0.02. A novel aspect of this study involves considering both passive and active control of the Maxwell nanofluid. The governing equations are solved using the Laplace transform technique to obtain uniform closed-form solutions. Graphical representations illustrate the velocity, temperature, concentration, and Bio-Convection profiles under various flow parameters. Remarkably, the velocity profile exhibits consistent behavior for both stationary η = 0 and moving η = 1 plates, as depicted in Figures 2–10. The obtained semi-analytical solutions successfully fulfill all initial and boundary conditions. Moreover, manipulating the nanoparticle volume fraction demonstrates significant control over the flow and heat transfer characteristics.

Application of constant proportional caputo (CPC) fractional derivative for natural convective casson nanofluid flow on an infinite cylinder

The article focuses on the application of fractional derivatives in the context of drilling fluids composed of clay nanoparticles (NPs) suspended in a Casson fluid. The main points of this research are composition of clay NPs with Casson fluid, mathematical approach (PDEs), flow processes, geometric and temperature conditions, Constant Proportional Caputo (CPC) fractional derivatives, analytical solution, verification through Mathcad, effect of volume fraction, thermal conductivity analysis, comparison of fractional derivatives, and accuracy of results. This article discusses the application of hybrid fractional derivatives in analyzing heat transmission in nanofluid drilling, particularly focusing on clay NP suspensions in a Casson fluid. The study aims to provide analytical solutions for temperature and velocity distributions by transformed into new hybrid fractional derivatives (CPC) and the fluid dynamics are influenced by the circular geometry and temperature changes. While also investigating the impact of different parameters and comparing the accuracy of different fractional derivative approaches.

Application of fractional derivatives in a Darcy medium natural convection flow of MHD nanofluid

Application of fractional derivatives in a Darcy medium natural convection flow of MHD nanofluid

Conformable finite element method for conformable fractional partial differential equations

<abstract><p>The finite element (FE) method is a widely used numerical technique for approximating solutions to various problems in different fields such as thermal diffusion, mechanics of continuous media, electromagnetism and multi-physics problems. Recently, there has been growing interest among researchers in the application of fractional derivatives. In this paper, we present a generalization of the FE method known as the conformable finite element method, which is specifically designed to solve conformable fractional partial differential equations (CF-PDE). We introduce the basis functions that are used to approximate the solution of CF-PDE and provide error estimation techniques. Furthermore, we provide an illustrative example to demonstrate the effectiveness of the proposed method. This work serves as a starting point for tackling more complex problems involving fractional derivatives.</p></abstract>

Application of Fractional Derivatives in Characterization of ECG graphs

In this paper, we have discussed the prediction of Left Ventricular Hypertrophy (LVH) and Right Ventricular Hypertrophy (RVH) of the heart from an Electrocardiogram (ECG) Graph using fractional order calculus. An ECG is a rough or unreachable curve that is continuous everywhere but non-differentiable at some points or all points where classical calculus fails. The purpose of this paper is to find left and right fractional derivatives at those non-differentiable points and then predict LVH and RVH by calculating the phase transition values (absolute difference of left and right fractional derivatives). Fractal dimension and Hurst exponents of V1, V2, V5, and V6 leads of the ECGs have been calculated for both problematic and normal ECGs. All such measures will help doctors to diagnose LVH and RVH from ECG in a more accurate manner as compared to the techniques which are under the classical order method.

Physical Interpretation of Nanofluid (Copper Oxide and Silver) with Slip and Mixed Convection Effects: Applications of Fractional Derivatives

A fractional model was developed for presenting the thermal assessment of nanoparticles in an inclined moving surface. Water was used as a base fluid, while the nanofluid utilized copper oxide and silver nanoparticles. The modification of the thermal model was further supported by mixed convection, magnetic force, and porous saturated space. Slip effects to the porous surface were also introduced. The fluctuation in temperature at different times was assumed by following the ramped thermal constraints. The fractional computations for the set of flow problems were performed with implementations of the Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) analytical techniques. The integration process for such computations was achieved using the Laplace transformation. The comparative velocity and thermal analysis for the water and kerosene-oil-based nanofluid model is presented. The declining change in the velocity was observed due to the increase in the volume fraction of nanoparticles. It was observed that the increment in the temperature profile was more progressive for the kerosene oil and silver nanoparticle suspension.

Applications of fractional derivatives in MHD free-convective oscillating flow of a blood based CNTs nanofluid across a porous medium

In this continuation, the mixed free convection of carbon nanotube (CNTs) is inspected in a by using the fractional approach. As a base fluid, human blood is considered and carbon nanotube (CNTs) (single and multi-wall carbon nanotubes) are detached in the form of blood-CNTs nanofluid. The leading partial derivative equations of the problem are non dimensionalized with appropriate non-dimensional variables. The leading equations are transformed into the fractional model with modified explanations and formulas of fractional framework. On this end, the Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) definitions have been followed. To find the semi-analytical solution of heat and momentum equations of the fractional model, the Laplace transformation technique is operated via MATLAB software. The comparison between CF and AB fractional algorithms is worked out to judge the accuracy of solution.

Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects

Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

Viscoelastic Effects on the Response of Electroelastic Materials.

Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or generators in some developments and applications. This is due to their capacity of being deformed when submitted to an electric field. Some models of their actuation are available, but recently, viscoelastic models have been proposed to give an account of the dissipative behaviour of these materials. Their response to an external mechanical or electrical force field implies a relaxation process towards a new state of thermodynamic equilibrium, which can be described by a relaxation time. However, it is well known that viscoelastic and dielectric materials, as for example, polymers, exhibit a distribution of relaxation times instead of a single relaxation time. In the present approach, a continuous distribution of relaxation times is proposed via the introduction of fractional derivatives of the stress and strain, which gives a better account of the material behaviour. The application of fractional derivatives is described and a comparison with former results is made. Then, a double generalisation is carried out: the first one is referred to the viscoelastic or dielectric models and is addressed to obtain a nonsymmetric spectrum of relaxation times, and the second one is the adoption of the more realistic Mooney–Rivlin equation for the stress–strain relationship of the elastomeric material. A modified Mooney–Rivlin model for the free energy density of a hyperelastic material, VHB 4910 has been used based on experimental results of previous authors. This last proposal ensures the appearance of the bifurcation phenomena which is analysed for equibiaxial dead loads; time-dependent bifurcation phenomena are predicted by the extended Mooney–Rivlin equations.

New Extension of Fractional-Calculus Results Associated with Product of Certain Special Functions

Fractional order calculus and special functions play a great role in scientific, financial and technological fields. In view of considerable impact and applications of fractional derivatives and integrals in real life, we aim to suggest some main formulas for the product of generalized M-series, $${\overline{\text{H}}}$$ -function and Aleph function associated with the Riemann-Liouvillle, the Weyl and many other operators of fractional order, which are derived by using the concept of the Cauchy-Goursat integral formula. The formulas derived in the present study can be employed to examine a broad class of new and known formulas involving simpler special functions, hitherto scattered in the literature.

Ferromagnetic coil frequency response and dynamics modeling with fractional elements

The modeling of coils with ferromagnetic cores is considered. An attempt is made at the application of fractional derivatives because of the success in the modeling of real phenomena in various other fields. Three coils with ferromagnetic cores have been selected for the analysis. In order to obtain a basis for the study, measurements have been taken on a specially designed setup including an arbitrary function generator and a data acquisition device. The measurement process has been controlled by an original algorithm written in C#. The studied model has been considered in several varieties (various numbers of fractional coil and capacitor branches) in a ladder-like structure. The estimation process has been performed in an original program written in C#, applying the COBYLA method for constrained optimization. The modeling and simulation results have been compared through their frequency characteristics. A more critical approach has also been applied, where the measurements and simulations are compared for both a non-sinusoidal waveform and a step function on the source output.

Application of fractional derivatives for the description of kinetic curvature of thermooxidation destruction of the polymers (Part I)

Application of fractional derivatives for the description of kinetic curvature of thermooxidation destruction of the polymers (Part I)

Application of fractional derivatives for description of kinetic curvature of thermooxylic polymers (part II)

Application of fractional derivatives for description of kinetic curvature of thermooxylic polymers (part II)

Estimation of soil salt content using machine learning techniques based on remote-sensing fractional derivatives, a case study in the Ebinur Lake Wetland National Nature Reserve, Northwest China

AbstractSoil salinity is a common global environmental problem that severely restricts industrial, agricultural and human development. In Northwest China, soil salinity is a problem affecting the Lake Ebinur area and needs to be monitored and addressed. The use of optical remote sensing technology to for timely and accurate soil salinity monitoring has great potentials and can be crucial for industrial and agricultural development. Optical remote sensing technology is an important data source for monitoring of soil salinization because of its rich spectral information and high-resolution. Based on the HJ-1/HSI data of fractional derivative transformation, this paper calculated the parameters, such as the difference index (DI), normalized difference index (NDI), ratio index (RI) and salt index (SI), that can invert the spatial distribution of salinization and then calculates a particle swarm optimization support vector machine (PSO-SVM). These index parameters were used to invert the spatial distribution information of salinized soil to explore the influence of fractional derivatives on the estimation accuracy of the spatial distribution of salinized soil. The results showed the following: (1) The optimal model based on different fractional derivatives and exponential transformations was the DI transformation under the fractional derivative of 1.2-order. R2, RMSE and RPD were 0.66, 13.81 g/kg and 2.59, respectively. These data show that the model had a good prediction effect. (2) The application of fractional derivatives in the processing of hyperspectral images could highlight the spectral details of the images and enrich their pretreatment methods, which is of great significance for the inversion of the distribution of salinized soil by hyperspectral images. (3) In the lakeside area of Lake Ebinur, the soil is mainly saline to severely saline. In the northern mountainous region of Lake Ebinur, the soil is mainly non-saline soil to mildly saline soil. Finally, in the Kuitun riverbank area in the southeast of Lake Ebinur, the soils are mainly non-saline soils to mildly saline soils. The results of this study demonstrate the potential of spectral index optimization and prediction models to monitor soil salinization based on fractional derivatives. This technique highlights the spectral detail information and enriches the preprocessing methods of hyperspectral imagery, resulting in the mapping of the spatial distribution of the degree of soil salinization. This has practical importance for the prevention and control of soil salinization and its spatial extent in arid and semi-arid regions.

An application of fractional derivatives to a thermo-convective viscous fluid with Dufour and Schmidt effects over a rotating disk

An application of fractional derivatives to a thermo-convective viscous fluid with Dufour and Schmidt effects over a rotating disk

Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients

In this work, a new fourth order compact approximation is derived for Riemann–Liouville space fractional derivatives. Modified wave numbers are obtained for various approximations of fractional derivatives. Considering these modified wave numbers, Fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations. A novel fourth order compact scheme is developed for space fractional advection–diffusion reaction equations with variable coefficients in one and two dimensions. In the proposed compact scheme, the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations. This splitting of second derivative approximations allows us to obtain a similar system of linear equations as in Zhao and Tian (2017) which presents only second order accurate finite difference method for advection–diffusion reaction equations in one-dimension. The stability of the proposed compact scheme is demonstrated numerically. Furthermore, the proposed compact scheme is employed to space fractional Black–Scholes equation for pricing European options as an application of fractional derivatives in mathematical finance. Numerical illustrations are presented to validate the theoretical claims.

Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate

Nanofluids are a novel class of heat transfer fluid that plays a vital role in industries. In mathematical investigations, these fluids are modeled in terms of traditional integer-order partial differential equations (PDEs). It is recognized that traditional PDEs cannot decode the complex behavior of physical flow parameters and memory effects. Therefore, this article intends to study the mixed convection heat transfer in nanofluid over an inclined vertical plate via fractional derivatives approach. The problem in hand is modeled in connection with Atangana–Baleanu fractional derivatives without singular and local kernel with a strong memory. Human blood is considered as base fluid and carbon nanotube (CNTs) (single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs)) are dispersed into it to form blood-CNTs nanofluid. The nanofluid is considered to flow in a saturated porous medium under the influence of an applied magnetic field. The exact analytical expressions for velocity and temperature profiles are acquired using the Laplace transform technique and plotted in various graphs. The empirical results indicate that the memory effect decreases with increasing fractional parameters in the case of both temperature and velocity profiles. Moreover, the temperature profile is higher for blood SWCNTs because of higher thermal conductivity whereas this trend is found opposite in the case of velocity profile due to densities difference.

Application of fractional derivative on non-linear biochemical reaction models

Systems of ordinary differential equations play an important role in analyzing the dynamics for real world situations such as cell signaling pathways, population growth, enzymatic inhibitor reactions, and ecological models. Although, using differential equations have a great advantage to understand the dynamical behavior for such systems but most of the biological models have memory or sometimes called after effects. Such effects in the systems are often neglected. The idea of fractional-order differential equations gives a great role in understanding and identifying these effects on the model dynamics. In this paper, we review the basic ideas of fractional differential equations and their applications on non-linear biochemical reaction models. We apply this idea to a non-linear model of enzyme inhibitor reactions. The suggested method provides a good step forwards to understand the model dynamics in complex enzymatic reactions. We use a numerical approach to calculate some computational simulation of the model for different initial conditions and parameters.