Fractional Differential Operator Research Articles

A Comparative Study on Solution Methods for Fractional order Delay Differential Equations and its Applications

Fractional calculus is one of the evolving fields in applied sciences. Delay differential equation of non-integer order plays a vital role in epidemiology, population growth, physiology economy, medicine, chemistry, control, and electrodynamics and many mathematical modeling problems Fractional Delay differential equations usually lacks analytic solutions and some of these equations can only be solved by some numerical methods. In this review article we present a comparative study on some standard numerical methods applied to solve linear fractional order differential equations with time delay. Fractional finite difference method (FFDM), Predictor-corrector method (PCM) with new and extended versions has been discussed in this article. All above mentioned methods use the Caputo type fractional differential operator to define fractional derivatives. Solution of a real-life problem formulated by FDDEs has been discussed under these methods. Results have been presented in tabular and graphical form to analyze the efficiency and scarcity of mentioned methods. These graphical and numerical comparisons are provided to illustrate and corroborate the similarity and differences between these methods.

Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $$\mathcal {D}_t^\alpha (\partial _{t}u)$$ and $$\mathcal {D}_t^\alpha u $$ (with $$\alpha \in (0,1)$$ ), where $$\mathcal {D}_t^\alpha $$ denotes the Caputo fractional differential operator (in time) of order $$\alpha $$ . We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe’s method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multivariate Mittag-Leffler function.

Scale-Invariant General Fractional Calculus: Mellin Convolution Operators

General fractional calculus (GFC) of operators that is defined through the Mellin convolution instead of Laplace convolution is proposed. This calculus of Mellin convolution operators can be considered as an analogue of the Luchko GFC for the Laplace convolution operators. The proposed general fractional differential operators are generalizations of scaling (dilation) differential operator for the case of general form of nonlocality. Semi-group and scale-invariant properties of these operators are proven. The Hadamard and Hadamard-type fractional operators are special case of the proposed operators. The fundamental theorems for the scale-invariant general fractional operators are proven. The proposed GFC can be applied in the study of dynamics, which is characterized by nonlocality and scale invariance.

Modeling and complexity analysis of a fractional-order memristor conservative chaotic system

Memristors are often utilized in circuit model analysis as one of the fundamental circuit components. In this paper, a five-dimensional conservative memristor chaotic system is built after the introduction of the memristor into a four-dimensional conservative chaotic system. The dynamic changes of the system are examined using phase diagram, mean value, and Lyapunov exponent spectrum. A line equilibrium point, symmetry and multi-stability are characteristics of the system; the phase trajectory can also produce shrinking and structure transformation behavior with the change of parameters. Furthermore, the system has initial offset boosting behaviors, conservative flows of it can be altered in position by changing two initial values, respectively. Most notably, we discover that the complexity of the system rises with the inclusion of memristor and again with the addition of fractional differential operators. It is shown that the complexity of chaotic systems may increase with the addition of memristors and fractional-order differential operators. At last, the NIST is used to test the randomness of the sequence, and the system's physical realizability is confirmed by the DSP platform.

Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups

We establish a novel definition of the fractional gradient and divergence of order $$\alpha \in (0, 1)$$ through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and demonstrate their blow-up property, using similar methods as outlined in [13].

Nonlocal (In Time) Problem for the Evolutionary Equation with Fractional Differential Operator

Nonlocal (In Time) Problem for the Evolutionary Equation with Fractional Differential Operator

Inverse Sturm-Liouville problem with conformable derivative and transmission conditions

In this paper, we study the inverse problem for Sturm-Liouville problem with conformable fractional differential operators of order $\alpha$, $0.5 < \alpha\leq 1$ and finite number of interior discontinuous conditions. For this aim first, the asymptotic formulas for solutions, eigenvalues and eigenfunctions of the problem are calculated. Then some uniqueness theorems for proposed inverse eigenvalue problem are proved. Finally, the Hald's theorem for conformable Sturm-Liouville problem is developed.

A Novel Fractional Sine Chaotic Map and Its Application to Image Encryption and Watermarking

This article addresses the telecommunications industry’s priority of ensuring information security during the transition to next-generation networks. It proposes an image encryption system that combines watermarking techniques and a discrete fractional sine chaotic map. The authors also incorporate the principles of blockchain to enhance the security of transmitted and received image data. The proposed system utilizes a newly developed sine chaotic map with a fractional difference operator, exhibiting long-term chaotic dynamics. The complexity of this map is demonstrated by comparing it with three other fractional chaotic maps from existing literature, using bifurcation diagrams and the largest Lyapunov exponent. The authors also show the map’s sensitivity to changes in initial conditions through time-series diagrams. To encrypt images, the authors suggest a method involving watermarking of two secret images and encryption based on blockchain technology. The cover image is watermarked with the two hidden images using discrete wavelet transformations. Then, the image pixels undergo diffusion using a chaotic matrix generated from the discrete fractional sine chaotic map. This encryption process aims to protect the image data and make it resistant to unauthorized access. To evaluate the algorithm, the authors perform statistical analysis and critical sensitivity analysis to examine its characteristics. They also analyse different attacks to assess the algorithm’s ability to resist such threats and maintain image quality after decryption. The results demonstrate that the proposed algorithm effectively defends against attacks and ensures image security.

Anomalous Relaxation and Three-Level System: A Fractional Schrödinger Equation Approach

We investigate a three-level system in the context of the fractional Schrödinger equation by considering fractional differential operators in time and space, which promote anomalous relaxations and spreading of the wave packet. We first consider the three-level system omitting the kinetic term, i.e., taking into account only the transition among the levels, to analyze the effect of the fractional time derivative. Afterward, we incorporate a kinetic term and the fractional derivative in space to analyze simultaneous wave packet transition and spreading among the levels. For these cases, we obtain analytical and numerical solutions. Our results show a wide variety of behaviors connected to the fractional operators, such as the non-conservation of probability and the anomalous spread of the wave packet.

Modelling and analysis of bad impact of smoking in society with Constant Proportional-Caputo Fabrizio operator

Smoking is now regarded as an epidemic because of the excessive mortality rate and large costs related to it. In line with the arena of the health enterprize, smoking is the third most common cause of mortality among human beings and the most preventable cause of sickness. Intending to study this problem, this manuscript frequently stresses the usage of the hybrid fractional differential operator to evaluate the dynamics of the fractional order quitting smoking version. We study a fractional-order version with a Constant Proportional-Caputo Fabrizio (CPCF) operator to investigate and examine the dynamical transmission of smoking. The qualitative analysis, and well-posedness of the fractional order system are examined. The analysis of the Volterra-type Lyapunov function for global stability is verified with the first and derivative tests. Also, Banach contraction characteristics are used to fulfil the criteria for the uniqueness of the exact solution. A complete discussion of additional analysis of the CPCF operator is furnished. In order to demonstrate the impacts of modifying the fractional order and to validate the theoretical findings, numerical simulations are carried out using an iterative Laplace transform method for a variety of fractional orders. The stability of the iterative scheme is also verified by using Picard’s stable condition from the fixed point theorem. The CPCF operator has a broader definition than the Caputo–Fabrizio (CF) operator, and it elaborates on new insights into the mechanisms of smoking transmission throughout society.

A Mini-Review on Recent Fractional Models for Agri-Food Problems

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India.

Analysis of the Solution of a Model of SARS-CoV-2 Variants and Its Approximation Using Two-Step Lagrange Polynomial and Euler Techniques

In this paper, a vaccination model for SARS-CoV-2 variants is proposed and is studied using fractional differential operators involving a non-singular kernel. It is worth mentioning that variability in transmission rates occurs because of the particular population that is vaccinated, and hence, the asymptomatic infected classes are classified on the basis of their vaccination history. Using the Banach contraction principle and the Arzela–Ascoli theorem, existence and uniqueness results for the proposed model are presented. Two different numerical approaches, the fractional Euler and Lagrange polynomial methods, are employed to approximate the model’s solution. The model is then fitted to data associated with COVID-19 deaths in Pakistan between 1 January 2022 and 10 April 2022. It is concluded that our model is much aligned with the data when the order of the fractional derivative ζ=0.96. The two different approaches are then compared with different step sizes. It is observed that they behave alike for small step sizes and exhibit different behaviour for larger step sizes. Based on the numerical assessment of the model presented herein, the impact of vaccination and the fractional order are highlighted. It is also noted that vaccination could remarkably decrease the spikes of different emerging variants of SARS-CoV-2 within the population.

A novel HIV model through fractional enlarged integral and differential operators

This article presents a novel mathematical fractional model to examine the transmission of HIV. The new HIV model is built using recently fractional enlarged differential and integral operators. The existence and uniqueness findings for the suggested fractional HIV model are investigated using Leray–Schauder nonlinear alternative (LSNA) and Banach’s fixed point (BFP) theorems. Furthermore, multiple types of Ulam stability (U-S) are created for the fractional model of HIV. It is straightforward to identify that the gained findings may be decreased to many results obtained in former works of literature.

On Novel Fractional Integral and Differential Operators and Their Properties

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

An improved FIGARCH model with the fractional differencing operator (1-νL)d

In this study, we propose a new volatility model called the product memory GARCH model by replacing (1 − L)d in the FIGARCH model of Baillie et al. (1996) with (1 − νL)d for 0 ≤ ν ≤ 1, where the impulse response function in its ARCH(∞) process has a decay rate in the form of the product of a geometric memory rate and a hyperbolic memory rate when the lags increase. Under the ARCH(∞) framework, the non-negativity constraints, the existence of the second, fourth, sixth, and eighth moments, and the memory length are investigated, and the Gaussian quasi-maximum likelihood estimation is also discussed. This improved model is of considerable significance in the evolution of GARCH-type models and financial econometrics.

On discrete tempered fractional calculus and its application

Discrete tempered fractional calculus, as a fresh generalization of discrete fractional calculus, takes both the advantages of discrete fractional calculus and its tempered fractional counterpart, which has great potential value in the interpretation of special physical phenomena. However, its essential characteristics are far from clarifying. Meanwhile, the primary goal of this study is to enrich the vital characteristics of discrete tempered fractional calculus in the presence of required domains for diverse scenarios. First, the reciprocal properties of the tempered fractional summation and difference operators, as well as the discrete version of Taylor’s formula, are established. The formulae of the discrete Laplace transform for discrete tempered fractional calculus are also determined with the aid of discrete convolution. Then, the well-posedness of tempered fractional difference equation is investigated, along with the exploration on the corresponding Volterra summation equation, and the existence, uniqueness and continuous dependence as well. In the aspect of application, the criterion of Ulam–Hyers stability of tempered fractional difference equation on finite time interval is obtained in light of a novel discrete Gronwall inequality. Besides, all examples are showed to verify the effectiveness of the main results.

Analytical solutions of fractional couple stress fluid flow for an engineering problem

Abstract In this article, analytical solutions of couple stress fluid flow modeled with a power law fractional differential operator are discussed. Stokes’ second problem for an incompressible couple stress fluid is studied for an horizontal plate of infinite length. The governing equations of the flow problem are expressed in terms of a partial differential operator and then converted into a non-dimensional model by using dimensional analysis. Then the integer order problem was formulated in terms of the non-integer order of three types of fractional derivatives and then solved with the help of the Laplace transform method. The obtained solutions are complex and expressed in terms of series. In order to check the memory index of the solutions obtained with three different fractional operators, we have plotted some graphs. It is found that the constant proportional operator provides us a better choice about the memory and maximum enhancement achieved in the comparison of Caputo and Caputo–Fabrizio. Furthermore, in order to check the accuracy of the present results, we have compared the obtained solutions with the existing literature and found a good agreement between them.

Impact of nanoparticles on vegetable oil as a cutting fluid with fractional ramped analysis

Better electrical insulation and thermal properties of vegetable oil with nanoparticles are crucial for its uses as a replacement for conventional previous lubricants used in heavy and light industries for cutting and machining. In this study, a magnetohydrodynamic (MHD) flow of a Brinkman-type nanofluid is used to investigate an infinite vertical plate with chemical reaction, heat radiation, and MHD flow. In order to improve the machining and cutting powers of regular vegetable oil, four distinct types of nanoparticles were selected to be the base fluid. The problem is modeled by coupled system partial differential equations (PDEs), and the results are generalized by the Caputo-Fabrizio fractional differential operator for the exponential non-singular kernel. In order to prepare nanofluids, four different types of nanoparticles, namely graphene oxide (GO), molybdenum disulfide (MoS2), titanium dioxide (TiO2), and aluminum oxide (Al2O3) are suspended separately in vegetable oil. The results of skin friction, the Nusselt number, and the Sherwood number are computed in various tables. It is found that GO nanoparticles, (followed by MoS2, TiO2, and Al2O3) are the materials that can heat transfer at the maximum rate. The heat transfer rate for GO is found to be the greatest with an enhancement up to 19.83% when 4% of nanoparticles are dispersed, followed by molybdenum disulfide at 16.96%, titanium dioxide at 16.25%, and alumina at 15.80%.

Combined Liouville–Caputo Fractional Differential Equation

This paper studies a fractional differential equation combined with a Liouville–Caputo fractional differential operator, namely, LCDηβ,γQ(t)=λϑ(t,Q(t)),t∈[c,d],β,γ∈(0,1],η∈[0,1], where Q(c)=qc is a bounded and non-negative initial value. The function ϑ:[c,d]×R→R is Lipschitz continuous in the second variable, λ>0 is a constant and the operator LCDηβ,γ is a convex combination of the left and the right Liouville–Caputo fractional derivatives. We study the well-posedness using the fixed-point theorem, estimate the growth bounds of the solution and examine the asymptotic behaviours of the solutions. Our findings are illustrated with some analytical and numerical examples. Furthermore, we investigate the effect of noise on the growth behaviour of the solution to the combined Liouville–Caputo fractional differential equation.

Fractional Schrödinger equation and time dependent potentials

We investigate the solutions for a time-dependent potential by considering two scenarios for the fractional Schrödinger equation. The first scenario analyzes the influence of the time-dependent potential in the absence of the kinetic term. We obtain analytical and numerical solutions for this case by considering the Caputo fractional time derivative, which extends Rabi’s model. In the second scenario, we incorporate the kinetic term in the Schrödinger equation and consider fractional spatial derivatives. For this case, we analyze the spreading of the Gaussian wave package under the action of the time and spatial fractional differential operators.