Fractional Differential Operator Research Articles

Analytical and fractional model for power transmission of lossy transmission line

ABSTRACT A lossy transmission line can draw current from DC source if DC voltage is applied to constant resistance. This manuscript proposes a fractional modeling of lossy transmission line based on the partial differential equations by employing Kirchoff’s current and voltage laws via Fourier analysis. The governing equation of lossy transmission line is fractionalized by means of modern fractional differential operators. The optimal solution of voltage is investigated by means of Fourier sine and Laplace transforms subject to the imposed conditions. The solutions of voltage over the transmission line have been investigated in terms of exponential and gamma functions. The comparative analysis of voltage over the transmission line through Caputo-Fabrizio and Atangana-Baleanu fractional operators has been presented for line losses on the basis of conductance, resistance and inductance for power transmission.

Application of Newton’s polynomial interpolation scheme for variable order fractional derivative with power-law kernel

This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo–Fabrizio derivative, the Atangana–Baleanu fractal and fractional derivative, and the Atangana–Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton’s polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang–Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach’s significance and effectiveness through numerical examples.

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems.

A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors.

Variable order fractional diabetes models: numerical treatment

ABSTRACT In this paper, we present two numerical approximation solutions for the updated nonlinear variable order fractional differential equations of diabetes disease. The variable order fractional differential operator is based on the Caputo fractional derivative concept. The first used method is developed using the nonstandard finite difference process, such that a new quick approximation is applied to tackle the nonlinear terms. By applying some reasonable assumptions to the given data, we can prove that this scheme maintains the positivity and boundedness of the discretized solutions. The second method is based on approximation with shifted Jacobi polynomials. The properties of Jacobi polynomials, together with the shifted Jacobi-Gauss-Lobatto nodes were utilized to reduce the proposed system into a set of algebraic nonlinear equations. Furthermore, some numerical experiments are conducted to compare the performance of the introduced schemes. Our numerical simulations show that the nonstandard finite difference approach requires less computational time compared to the Jacobi-Gauss-Lobatto spectral collocation. While the last one is more accurate when used for solving the variable order fractional diabetes model.

Applications of fractional difference operators for new version of Brudno-Mazur Orlicz bounded consistency theorem

In this paper, we intend to prove that the modulus $\mathcal{A}-$lacunary statistical convergence of fractional difference double sequences and modulus lacunary fractional matrix of four-dimensions taken over the space of modulus $\mathcal{A}-$lacunary fractional difference uniformly integrable real sequences are equivalent. We represent another version of the Brudno-Mazur Orlicz bounded consistency theorem by using modulus function, lacunary sequence, and fractional difference operator. We show that the four-dimensional $RH-$ regular matrices $\mathcal{A}$ and $\mathcal{B}$ are modulus lacunary fractional difference consistent over the multipliers space of modulus fractional difference $\mathcal{A}-$summable sequences and an algebra $Z.$

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ-Hilfer, fractional derivative of order μ∈(1,2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ-Laplace transform, w-weighted ψ-convolution and the measure of non-compactness. Since the operator, the w-weighted Φ-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.

An analysis of exponential kernel fractional difference operator for delta positivity

Abstract Positivity analysis for a fractional difference operator including an exponential formula in its kernel has been examined. A composition of two fractional difference operators of order ( ν , μ ) \left(\nu ,\mu ) in the sense of Liouville–Caputo type operators has been analysed in cases when ν ≠ μ \nu \ne \mu and ν = μ \nu =\mu . Due to the kernel of the fractional difference operator being convergent, there has been a restriction in the domain of the solution. Incidentally, a negative lower bounded condition has been carried out through analysing the positivity results. For a better understanding, an increasing function has been considered as a test for the main results.

Investigating integrodifferential equations associated with fractal–fractional differential operators

This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach.

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis

In light of fractional theory, this paper presents several new effective solitonic formulations for the Langmuir and ion sound wave equations. Prior to this study, no previous research has presented the comparision and obtained the generalized fractional soliton solutions of this kind with power law kernel and Mittag-Leffler kernel. The ion sound and Langmuir wave equations are essential in plasma physics, offering insights into the collective behavior of charged particles in plasmas and enabling diagnostics and control of these complex, ionized gas systems. The two distinct fractional order differential operators are substituted for the traditional order derivative to reshape the examined model. The Atangana-Baleanu non-singular and non-local operator and conformable fractional operator are the fractional-order operators that are used to create the fractional complex system equations for Langmuir waves and ion sound. A constructive approach new auxiliary equation method utilizes to obtain the exact analytical soliton solutions for ion sound and Langmuir wave equation. A wide range of soliton solutions is obtained, including mixed complex solitary shock solutions, singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed singular solutions, exact solutions, mixed periodic solutions, and mixed hyperbolic solutions, dark soliton, bright soliton, trigonometric solutions, periodic results, and hyperbolic results. The solitons solution of the ion sound and Langmuir wave equations lies in their ability to maintain wave stability, their role in modeling wave propagation and nonlinear effects, their potential use as diagnostic tools, and their relevance in wave-particle interactions in plasma physics. The solitons provide a valuable framework for understanding the behavior of waves in plasmas and offer insights into the complex dynamics of these charged particle systems. A graphical comparison analysis of a few solutions is also shown here, taking into account appropriate parametric values through the use of the software package. Moreover, the results of this study have important implications for Hamilton's equations and generalized momentum, where solitons are employed in long-range interactions.

APPLICATION OF THE SAMUELSON EQUATION TO THE EVANS MODEL

Mathematical modeling of economic processes is an actual direction of research, because the well-being of citizens and the country as a whole depends on it. In the case of the market, the prices of most goods and services are not planned centrally, are not directly regulated by the state, but are freely set and changed by the market itself. The main factors that control the movement of prices in the market are the demand and supply of goods. In the economy, the most important are dynamic models, the parameters of which change over time. The Evans model (Walras-Evans-Samuelson model) is currently one of the basic concepts explaining the dynamic establishment of the equilibrium price in the market of one product under the influence of supply and demand. This is due to the fact that knowing the dynamics of the economic parameter we are interested in, we can try to build a forecast of its further evolution. The article examines the market of one product. For convenience, we will assume that the functions of the dependence of demand and supply on the price are given by linear relationships. The construction of the Evans model is that the change in price is directly proportional to the excess of demand over supply and the duration of this excess. The Samuelson differential equation with an initial condition, that is, the Cauchy problem, is obtained. Samuelson's equation has a stationary (equilibrium) point, which is a positive price at which supply and demand will be equal. The analysis of the obtained solution of the problem shows that over a long enough time (relatively speaking, at ) the price asymptotically approaches the equilibrium value. If we are not interested in the temporal dependence, but only in the equilibrium price, then it can be found from the differential equation immediately by setting the condition , this is the so-called limit stationary mode. The solution of the Cauchy problem is found by the method of variation of the constant. The parameterization of the model using the fractional differentiation operator in the sense of Gerasimov-Caputo is considered. The resulting solution was analyzed depending on the parameter . For this, the asymptotic representation of the function was used for large values of the argument. Conclusions are made regarding price dynamics in time relative to the equilibrium price when demand and supply are equal.

The first boundary value problem for the fractional diffusion equation in a degenerate angular domain

This article addresses the problems observed in branching fractal structures, where super-slow transport processes can occur, a phenomenon described by diffusion equations with a fractional time derivative. The characteristic feature of these processes is their extremely slow relaxation rate, where a physical quantity changes more gradually than its first derivative. Such phenomena are sometimes categorized as processes with “residual memory”. The study presents a solution to the first boundary problem in an angular domain degenerating into a point at the initial moment of time for a fractional diffusion equation with the RiemannLiouville fractional differentiation operator with respect to time. It establishes the existence theorem of the problem under investigation and constructs a solution representation. The need for understanding these super-slow processes and their impact on fractal structures is identified and justified. The paper demonstrates how these processes contribute to the broader understanding of fractional diffusion equations, proving the theorem’s existence and formulating a solution representation.

A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution’s existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.

Modeling of implicit multi term fractional delay differential equation: Application in pollutant dispersion problem

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances.

Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels

Young children frequently die from measles, which is a major global health concern. Despite being more prevalent in infants, pregnant women, and people with compromised immune systems, it can infect anyone. Novel fractional operators, the constant-proportional Caputo operator, and the constant-proportional Atangana–Baleanu operator are used to create a hybrid fractional order model that helps analyze the dynamic transmission of the measles virus. We assess the measles-free and endemic equilibrium, reproductive number, biological viability, boundedness, well-posedness, and positivity of the model. We apply the Banach contraction principle to verify the uniqueness of the system’s solutions. The proposed system is confirmed to be Ulam–Hyres stable by using fixed point theory results. The aforementioned operators are further analyzed, and the Laplace-Adomian decomposition method is used to numerically simulate the system of fractional differential equations. To support our findings, the outcomes are graphically displayed. The efficacy and memory impact of fractional operators are illustrated through comparisons. Based on fractional parameter values, the study determines important disease-control strategies and shows that vaccinations greatly reduce the spread of measles. By reducing the number of infected people, increasing vaccination coverage lowers the burden of disease on the general population.

Data Dependence and Existence and Uniqueness for Hilfer Nabla Fractional Difference Equations

In the present article, we establish sufficient conditions for the existence of a unique bounded solution using a prominent fixed-point theorem for the non-linear initial value problem involving the recently introduced Hilfer nabla fractional difference operator. where 0 <ℵ<1, 0 ≤ ß ≤1, ι =ℵ+ ß −ℵß and j : Nx × Rn →Rn. We also analyze the Ulam-Hyers stability of the considered problem and make some interesting observations on the dependence of its solutions on initial conditions and parameters. Finally, we conclude this article by constructing suitable examples to illustrate the applicability of established results.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.