Derivatives Of Non-integer Order Research Articles

Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives

In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.

Fractional viscoelastic models with Caputo generalized fractional derivative

This article focuses on fractional Maxwell model of viscoelastic materials, which are a generalization of classic Maxwell model to noninteger order derivatives. We present and discuss formulations of the fractional order viscoelastic model and give physical interpretations of the model by using viscoelastic functions. We apply the generalized Caputo fractional derivative to viscoelastic models, namely fractional Maxwell model, fractional Kelvin‐Voigt model, and fractional Zener model. The stress relaxation module and creep compliance for each model are derived analytically using generalized Caputo fractional derivative. We analyze effect of α and newly introduced parameter ρ in all these models. The result shows an effect on viscoelastic models using fractional operator.

Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator D_{t}^{alpha }, more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley’s generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted.

Linearized novel operational matrices-based scheme for classes of nonlinear time-space fractional unsteady problems in 2D

Finding analytical and semi-analytical solutions of two-dimensional nonlinear fractional-order problems arising in mathematical physics is a challenging task for research community. In this work, an innovative scheme is proposed based on the shifted Gegenbauer wavelets. For this purpose first, we introduce shifted Gegenbauer polynomials via suitable transformation. Then, we present three-dimensional shifted Gegenbauer wavelets using shifted Gegenbauer polynomials. We illustrate function approximation of three variables, for example, u(x,y,t) through shifted Gegenbauer wavelets. To compute the novel operational matrices of positive integer and non-integer order derivative of shifted Gegenbauer wavelets vector in one, two and three dimensionals piecewise functions are utilized. Moreover, we describe associated theorems to validate our newly proposed scheme mathematically. The proposed algorithm is innovative because of the incorporation of Picard iterative scheme to tackle highly nonlinear problems of fractional-order. The current computational scheme coverts a mathematical model to a system of linear algebraic equations that are easier to solve. To validate the accuracy, credibility, and reliability of the present method, we analyze various fractional-order problems, including Bloch–Torrey, Burgers, Schrödinger, Rayleigh–Stokes and sine-Gordon. We also conduct a detailed comparative study, which demonstrates that the proposed computational scheme is effective to find the analytical and semi-analytical solutions of the aforementioned problems. Moreover, the proposed computational method can be utilized to analyze the solutions of other higher dimensional nonlinear fractional or variable order problems of physical nature.

Thermography of ferromagnetic Walter's-B fluid through varying thermal stratification

Thermally induced damages (hypolimnion, metalimnion and epilimnion) are usually caused by thermal stratification; due to a fact that fluid flows at a low velocity with large temperature variances. In this manuscript, thermal stratification phenomenon is studied for unsteady flow of Walter's-B fluid subject to a consistent inclined magnetic field at an angle of inclination over the boundary of transverse xy-plane with fractional thermal transport. Non-singular and local fractional derivative is hauled in the constitutive equations for the thermal flux. Semi analytical solutions of the dimensionless temperature and velocity fields are recognized by using the Laplace inversion numerical algorithms Stehfest's and Tzou's. Temperature and rate of heat transfer for non-integer and integer order derivatives are computed. Finally, the influence of some flow parameters and fractional parameter α on the temperature and velocity fields are graphically underlined by using IMATHCAD software and discussed.

Mathematical modeling of pine wilt disease with Caputo fractional operator

• Fractional order model with Caputo operator provides more flexible outcomes than the corresponding integer order. • Sensitive parameters essential for the control of the infection. • Fractional operators gives more biologically feasible behavior about the dynamic of pinewilt disease. In this work, we investigate the transmission dynamics of pine wilt disease infection and developed a new model utilizing Caputo fractional-order derivative. Moreover, with the use of fixed point theorem, the existence and uniqueness of the pine wilt disease model are obtained under Caputo operator. Using forward normalized sensitivity index, we determine the most sensitive parameters essential for the control of the infection and the results show that, decreasing the values of contact rate of a susceptible vector with infected pine trees and progression rate play a significant role in controlling the spread of pine wilt disease infection. On the other hand, we obtain different numerical simulations results of the model using the appropriate parameter values. Hence, from the graphs, it can be concluded that Caputo fractional operator gives more biologically observable behavior of the proposed disease model thanks to the changed fractional order. Compared to the previously built integer order model, the non-integer order derivative provided more efficient and flexible information about the complexity of the model’s dynamics.

Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

Thermal transmittance and thermo-magnetization of unsteady free convection viscous fluid through non-singular differentiations

Convective stability of ferromagnetic fluids can provide an adequate heat transfer via conventional convection due to a temperature gradient. This manuscript presents an analytical study for thermo-magnetized convection by means of non-integer order derivatives is investigated in which the core object is to show how to enhance, invert and suppress the convection modes based on local verses non-local kernels. Mathematical modeling is developed for the sake of magnetization depends upon governing equations of velocity, concentration and temperature profiles. Fractional treatments are invoked based on Fourier analysis and Laplace transforms to the dynamical equations govern the thermomagnetic convection. The analytical solutions of velocity, concentration and temperature have been established in terms of Mittage-Leffler and elementary functions. In order to have the performance of higher or lower susceptibility from thermomagnetic convection, profiles of velocity, concentration and temperature have been depicted separately by means of Atangana-Baleanu and Caputo-Fabrizio fractional operators for highlighting the immense impacts on ferromagnetic fluid flow. For the sake of novel outcomes of this study, it is observed that velocity is decreasing function of magnetic field when an increment in the amount of magnetic constant kicks off the augmentation of the Lorentz strength.

Non-Linear Macroeconomic Models of Growth with Memory

In this article, two well-known standard models with continuous time, which are proposed by two Nobel laureates in economics, Robert M. Solow and Robert E. Lucas, are generalized. The continuous time standard models of economic growth do not account for memory effects. Mathematically, this is due to the fact that these models describe equations with derivatives of integer orders. These derivatives are determined by the properties of the function in an infinitely small neighborhood of the considered time. In this article, we proposed two non-linear models of economic growth with memory, for which equations are derived and solutions of these equations are obtained. In the differential equations of these models, instead of the derivative of integer order, fractional derivatives of non-integer order are used, which allow describing long memory with power-law fading. Exact solutions for these non-linear fractional differential equations are obtained. The purpose of this article is to study the influence of memory effects on the rate of economic growth using the proposed simple models with memory as examples. As the methods of this study, exact solutions of fractional differential equations of the proposed models are used. We prove that the effects of memory can significantly (several times) change the growth rate, when other parameters of the model are unchanged.

On Dynamical Systems with Nabla Half Derivative on Time Scales

This paper is devoted to study of dynamical systems involving nabla half derivative on an arbitrary time scale. We prove existence and uniqueness of the solution of such system supplied with a suitable initial condition. Both Riemann–Liouville and Caputo approaches to noninteger-order derivatives are covered. Under special conditions we present an explicit form of the solution involving a time scales analogue of Mittag–Leffler function. Also an algorithm for solving of such problems on isolated time scales is established. Moreover, we show that half power functions are positive and decreasing with respect to $$t-s$$ on an arbitrary time scale.

Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators

A new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense.

Experimental Assessment of Fractional-Order PDD1/2 Control of a Brushless DC Motor with Inertial Load

The application of Fractional Calculus to control mechatronic devices is a promising research area. The most common approach to Fractional-Order (FO) control design is the PIλDµ scheme, which adopts integrals and derivatives of non-integer order λ and µ. A different possible approach is to add FO terms to the PID control, instead of replacing integer order terms; for example, in the PDD1/2 scheme, the half-derivative term is added to the classical PD. In the present paper, by mainly focusing on the transitory behaviour, a comparison among PD, PDµ, and PDD1/2 control schemes is carried out, with reference to a real-world mechatronic implementation: a position-controlled rotor actuated by a DC brushless motor. While using a general non-dimensional approach, the three control schemes are first compared by continuous-time simulations, and tuning criteria are outlined. Afterwards, the effects of the discrete-time digital implementation of the controllers are investigated by both simulation and experimental tests. The results show how PDD1/2 is an effective and almost cost-free solution for improving the trajectory-tracking performance in position control of mechatronic devices, with limited computational burden and, consequently, easily implementable on most commercial motion control drives.

Dirac particle with memory: Proper time non-locality

A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described.

A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel

In this paper, we analyze the dynamical behavior of fish farm model related to Atangana–Baleanu derivative of arbitrary order. The model is constituted with the group of nonlinear differential equations having nutrients, fish and mussel. We have included discrete kind gestational delay of fish. The solution of fish farm model is determined by employing homotopy analysis transforms method (HATM). Existence of and uniqueness of solution are studied through Picard–Lindelof approach. The influence of order of new non-integer order derivative on nutrients, fish and mussel is discussed. The complete study reveals that the outer food supplies manage the behavior of the model. Moreover, to show the outcomes of the study, some numerical results are demonstrated through graphs.

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

Abstract We 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.

On the evolution of fractional diffusive waves

In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time derivatives in the above standard equations with pseudo-differential operators interpreted as derivatives of non integer order (nowadays misnamed as of fractional order) we are lead to generalized processes of diffusion that may be interpreted as slow diffusion and interpolating between diffusion and wave propagation. In mathematical physics we may refer these interpolating processes to as fractional diffusion-wave phenomena. In this work we analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem. In the firsrst case we get the fundamental solutions (or Green functions) of the problem whereas in the latter case the solutions are obtained by a space convolution of the Green function with the input function. In order to clarify the matter for the non-specialist readers, we briefly recall the basic and essential notions of the fractional calculus (the mathematical theory that regards the integration and differentiation of noninteger order) with a look at the history of this discipline.

On stability of improved conformable model for studying the dynamics of a malnutrition community

The health of an individual is directly connected with the quality of food that a person takes in. Especially, a mother’s health is very important in the perspective that a healthy mother will have a healthy baby. This paper is the study of a conformable model of malnutrition community. A modified version of the mathematical model for malnutrition community has been developed. Stability of the developed model has been checked and an investigation (with the help of fixed-point theory) on the existence and uniqueness of the solution of the model has also been made. Suitable plots have been displayed to show the hidden behavior of non-integer order derivative.

A Survey of Models of Ultraslow Diffusion in Heterogeneous Materials

Ultraslow diffusion is characterized by a logarithmic growth of the mean squared displacement (MSD) as a function of time. It occurs in complex arrangements of molecules, microbes, and many-body systems. This paper reviews mechanical models for ultraslow diffusion in heterogeneous media from both macroscopic and microscopic perspectives. Macroscopic models are typically formulated in terms of a diffusion equation that employs noninteger order derivatives (distributed order, structural, and comb models (CM)) or employs a diffusion coefficient that is a function of space or time. Microscopic models are usually based on the continuous time random walk (CTRW) theory, but use a weighted logarithmic function as the limiting formula of the waiting time density. The similarities and differences between these models are analyzed and compared with each other. The corresponding MSD in each case is tabulated and discussed from the perspectives of the underlying assumptions and of real-world applications in heterogeneous materials. It is noted that the CMs can be considered as a type of two-dimensional distributed order fractional derivative model (DFDM), and that the structural derivative models (SDMs) generalize the DFDMs. The heterogeneous diffusion process model (HDPM) with time-dependent diffusivity can be rewritten to a local structural derivative diffusion model mathematically. The ergodic properties, aging effect, and velocity autocorrelation for the ultraslow diffusion models are also briefly discussed.

The approximate solution of charged particle motion equations in oscillating magnetic fields using the local multiquadrics collocation method

The charged particle motion for certain configurations of oscillating magnetic fields can be simulated by a Volterra integro-differential equation of the second order with time-periodic coefficients. This paper investigates a simple and accurate scheme for computationally solving these types of integro-differential equations. To start the method, we first reduce the integro-differential equations to equivalent Volterra integral equations of the second kind. Subsequently, the solution of the mentioned Volterra integral equations is estimated by the collocation method based on the local multiquadrics formulated on scattered points. We also expand the proposed method to solve fractional integro-differential equations including non-integer order derivatives. Since the offered method does not need any mesh generations on the solution domain, it can be recognized as a meshless method. To demonstrate the reliability and efficiency of the new technique, several illustrative examples are given. Moreover, the numerical results confirm that the method developed in the current paper in comparison with the method based on the globally supported multiquadrics has much lesser volume computing.

Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics.