Caputo Fractional Derivative Of Order Research Articles

A survey on random fractional differential equations involving the generalized Caputo fractional-order derivative

This article discusses the preliminary results of a general form of random fractional differential equations (RFDEs) involving the generalized Caputo fractional derivative. The problem proposed here allows us to interpolate various types of RFDEs. By applying the fixed point theorem and the generalized Gronwall fractional integral inequality, the existence of a unique solution and the Hyers–Ulam stability of RFDEs are presented. Furthermore, a new technique called the Hybrid Laplace-like transform method (HLTM) is proposed to seek the solutions of RFDEs in the nonlinear form. Numerical examples are provided to verify the effectiveness of the proposed method.

Cauchy problems for the time-fractional degenerate diffusion equations

This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with $\partial^{\alpha}_{t}$ the Caputo fractional derivative of order $\alpha\in(0,1)$ in the variable t and time-degenerate diffusive coefficients $t^{\beta}$ with $\beta >1-\alpha$. The solutions of  Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative $\partial^{\alpha}_{t}$ of order $\alpha\in(0,1)$  in the variable $t$, are shown. In the "Problem statement and main results" section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function $E_{\alpha,m,l}(z)$ and applying the Fourier transform $F$ and inverse Fourier transform $\mathcal{F}^{-1}$. Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.&nbsp

Error analysis of a finite difference scheme on a modified graded mesh for a time-fractional diffusion equation

This paper is concerned with a finite difference scheme on a new modified graded mesh for a time fractional diffusion equation with a Caputo fractional derivative of order α∈(0,1). At first, the construction and some basic properties of this new modified graded mesh are investigated and on its basis the L1 scheme is applied to approximate the Caputo derivative. Meanwhile, the standard center finite difference scheme on a uniform mesh is used to discretize the diffusion term. Then, stability and convergence of the proposed scheme in the maximum norm are proved. The convergence result shows that on this modified graded mesh one attains an optimal 2−α rate for the L1 scheme. Finally, the presented theoretical results are supported by some numerical experiments.

Study of parabolic motion effects on fractional order simulation for unsteady MHD nanofluid flow through porous medium

This study focuses on the impact of parabolic motion on unsteady free convective MHD nanofluid flow through a permeable medium. Effects of chemical reactions, heat generation and thermal radiation are also considered. Using similarity transformation, the governing equations are transformed into dimensionless form and generalised using the Caputo fractional order derivative. For solving the dimensionless equations, a suitable Laplace transform technique is applied and obtained the analytical expression. To understand the physical aspects of these problems, approximation results are defined using MATLAB software and presented the effects of different physical parameters on concentration, velocity and temperature profiles through graphs. From the results, it is concluded that a magnetic field tends to diminish the motion of fluid whereas thermal radiation and heat generation tend to improve the momentum of flow. It is also seen that heat transfer process increases due to the increasing values of radiation and heat generation.

Analytical Investigation of the Heat Transfer Effects of Non-Newtonian Hybrid Nanofluid in MHD Flow Past an Upright Plate Using the Caputo Fractional Order Derivative

The objective of this paper is to examine the augmentation of the heat transfer rate utilizing graphene (Gr) and multi-walled carbon nanotubes (MWCNTs) as nanoparticles, and water as a host fluid in magnetohydrodynamics (MHD) flow through an upright plate using Caputo fractional derivatives with a Brinkman model on the convective Casson hybrid nanofluid flow. The performance of hybrid nanofluids is examined with various shapes of nanoparticles. The Caputo fractional derivative is utilized to describe the governing fractional partial differential equations with initial and boundary conditions on the flow model. Exact solutions are obtained for flow transport, temperature distribution besides that heat transfer rate and friction drag in terms of Mittag-Leffler function by using Fourier sine and Laplace techniques as hybrid methods. Further, we provided the limiting case solutions for classic partial differential equations on obtained governing fluid flow models. The influence of various physical parameters with different fractional orders are investigated on hybrid nanofluid’s fractional momentum and energy by plotting velocity and energy curves. Few of the findings suggest that fractional parameters have significant effect on flow parameters and that blade-shaped nanoparticles have a high heat transfer rate. The graphical results reveal that the Grashof number shows a symmetry effect in the case of cooling and heating the plate. Furthermore, the performance of hybrid nanofluid is considerably more effective with the Caputo-fractional derivatives rather than in the classic derivative approach.

Strong convergence rates for the approximation of a stochastic time-fractional Allen–Cahn equation

The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α∈(0,1), and a Riemann–Liouville fractional integral operator of order γ∈[0,1] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results.

EXTENSION OF HAAR WAVELET TECHNIQUES FOR MITTAG-LEFFLER TYPE FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

Fractional order integro-differential equation (FOIDE) of Fredholm type is considered in this paper. The mentioned equations have many applications in mathematical modeling of real world phenomenon like image and signal processing. Keeping the aforementioned importance, we study the considered problem from two different aspects which include the existence theory and computation of numerical approximate solution. FOIDEs have been investigated very well by using Caputo-type derivative for the existence theory and numerical solutions. But the mentioned problems have very rarely considered under the Mittage-Leffler-type derivative. Also, for FOIDE of Fredholm type under Mittage-Leffler-type derivative has not yet treated by using Haar wavelet (HW) method. The aforementioned derivative is non-singular and nonlocal in nature as compared to classical Caputo derivative of fractional order. In many cases, the nonsingular nature is helpful in numerical computation. Therefore, we develop the existence theory for the considered problem by using fixed point theory. Sufficient conditions are established which demonstrate the existence and uniqueness of solution to the proposed problem. Further on utilizing HW method, a numerical scheme is developed to compute the approximate solution. Various numerical examples are given to demonstrate the applicability of our results. Also, comparison between exact and numerical solution for various fractional orders in the considered examples is given. Numerical results are displayed graphically.

On fractional evolution equations with an extended ψ−fractional derivative

This manuscript aims to highlight the existence and uniqueness results for a class of fuzzy nonlinear fractional evolution equations. Our approach is based on the application of an extended ??Caputo fractional derivative of order q ?(0,1) valid on fuzzy functions paired with Banach contraction principle. As an example of application, we provide one at the end of this paper to show how the results can be used.

The existence of a solution for nonlinear fractional differential equations where nonlinear term depends on the fractional and first order derivative of an unknown function

In this paper, we consider the existence of solutions of the nonlinear fractional differential equation boundary-value problem D?* u(t) = f (t, u(t), u?(t), CD?u(t)), 0 < t < 1, 1 < ? < 2, 0 < ? ? 1, u(0) = A, u(1) = Bu(?), where 0 < ? < 1, A ? 0, B? > 1, D?* is the modified Caputo fractional derivative of order ?, CD? is the Caputo fractional derivative of order ?, and f is a function in C([0, 1] ? R ? R ? R). Existence results for a solution are obtained. Two examples are presented to illustrate the results.

Simulation of generalized time fractional Gardner equation utilizing in plasma physics for non-linear propagation of ion-acoustic waves

In this work, radial basis function collocation method (RBFCM) is implemented for generalized time fractional Gardner equation (GTFGE). The RBFCM is meshless and easy-to-implement in complex geometries and higher dimensions, therefore, it is highly demanding. In this work, the Caputo derivative of fractional order ? ? (0, 1] is used to approximate the first order time derivative whereas, Crank-Nicolson scheme is hired to approximate space derivatives. The numerical solutions are presented and discussed, which demonstrate that the method is effective and accurate.

Controllability and Observability Analysis of a Fractional-Order Neutral Pantograph System

In the recent past, a number of research articles have explored the stability, existence, and uniqueness of the solutions and controllability of dynamical systems with a fractional order (FO). Nevertheless, aside from the controllability and other dynamical aspects, very little attention has been given to the observability of FO dynamical systems. This paper formulates a novel type of FO delay system of the Pantograph type in the Caputo sense and explores its controllability and observability results. This research endeavor begins with the conversion of the proposed dynamical system into a fixed-point problem by utilizing Laplace transforms, the convolution of Laplace functions, and the Mittag–Leffler function (MLF). We then set out Gramian matrices for both the controllability and observability of the linear parts of our proposed dynamical system and prove that both the Gramian matrices are invertible, thus confirming the controllability and observability in a given domain. Considering the controllability and observability results of the linear part along with some other assumptions, we investigate the controllability and observability results related to the nonlinear system. The Banach contraction result, the fixed-point result of Schaefer, the MLF, and the Caputo FO derivative are used as the main tools for establishing these results. To establish the authenticity of the established results, we add two examples at the end of the manuscript.

FRACTIONAL CALCULUS OPERATORS–BLOCH–TORREY PARTIAL DIFFERENTIAL EQUATION–ARTIFICIAL NEURAL NETWORKS–COMPUTATIONAL COMPLEXITY MODELING OF THE MICRO–MACROSTRUCTURAL BRAIN TISSUES WITH DIFFUSION MRI SIGNAL PROCESSING AND NEURONAL MULTI-COMPONENTS

Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients’ measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues’ complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter [Formula: see text] Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional orders. The fractional-order degree of the proposed and developed model is applied in relation to their corresponding complexity degrees. Consequently, experimentation and observations from the simulation-driven FFNN (with BFGS Quasi-Newton equation) learning scheme applied to the Bloch–Torrey PDE–Caputo FOD with MLF Neuron model (named as FFNN–BTPDE–CFODMLF Neuron model) proposed in this study, are made. Thus, by investigating whether the mathematical models based on the accurate neuron geometry models obtained can be optimized by comparing the errors in order to define the order parameter and identify to what optimal extent the errors are in relation to the prediction results with a particular focus on the neuron model, we have been able to estimate and predict brain microstructure through DMRI, accentuating mathematical and medical contributions based on the exploitation and corroboration of powerful modeling as well as computational capabilities.

Fractionalizing, coupling and methods for the coupled system of two-dimensional heat diffusion models

<abstract><p>The present manuscript gives an overview of how two-dimensional heat diffusion models underwent a fractional transformation, system coupling as well as solution treatment. The governing diffusion models, which are endowed with Caputo's fractional-order derivatives in time $ t $, are suitably coupled using the (1) convection phenomenon, (2) interfacial coupling by considering the mechanism of a double-layered bar, and the (3) nonlinear coupling due to temperature-dependent thermal diffusivities. Semi-analytical and analytical methods are considered for the solution treatment. Moreover, we seek a computational environment to graphically illustrate the systems' response to different fractional orders in each case through the determined diffusional fields. Besides, we supply certain concluding notes at the end.</p></abstract>

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system. For the proposed study, we consider the model under the conformable fractional order derivative (CFOD). We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution. Indeed, Schauder and Banach fixed point theorems are utilized to prove our claim. Further, an algorithm for the approximate analytical solution to the proposed problem has been established. In this regard, the conformable fractional differential transform (CFDT) technique is used to compute the required results in the form of a series. Using Matlab-16, we simulate the series solution to illustrate our results graphically. Finally, a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

A Fractional Cross-Entropy Based οn Caputo Fractional-Order Derivative

A Fractional Cross-Entropy Based οn Caputo Fractional-Order Derivative

FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

The sine-Gordon equation has received attention since 1970s due to the existence of soliton solutions. The aforesaid equation has significant applications in the quantum field theory. The aforementioned problem has been treated by using various numerical and analytical techniques under the ordinary as well as fractional-order derivatives. The mentioned equation has been investigated under the usual Caputo fractional-order derivative. Since in some cases the nonsingular-type derivatives produce more significant results in the mathematical modelings of real-world nonlinear problems, therefore, the proposed problem is considered in this paper under the fractional-order case in the context of Atangana–Baleanu–Caputo (ABC) derivative for the analytical and approximate results. This fractional derivative has some useful properties involving Mittag-Leffler-type kernel that is nonlocal and nonsingular. Furthermore, Modified Homotopy Perturbation Method (MHPM) is utilized for the required approximate solution. We give appropriate examples depicting the sine-Gordon model. Also, we present our results for the approximate solution graphically to support all the results.

Some Compound Fractional Poisson Processes

In this paper, we introduce and study fractional versions of the Bell–Touchard process, the Poisson-logarithmic process and the generalized Pólya–Aeppli process. The state probabilities of these compound fractional Poisson processes solve a system of fractional differential equations that involves the Caputo fractional derivative of order 0<β<1. It is shown that these processes are limiting cases of a recently introduced process, namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property, etc., for these processes. Further, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes.

Vaccination effect on the dynamics of dengue disease transmission models in Nepal: A fractional derivative approach

Dengue is a vector-borne disease which is spreading rapidly around the world. It is one of the fastly growing public health problems in Nepal. Since 2004, dengue cases have been recorded in both tropical and subtropical regions of Nepal. There is no specific treatment for the dengue disease, which is brought on by the dengue virus. This study attempts the use of the Caputo fractional-order derivative to suggest SVEIRP-SEI epidemic model that integrates vaccination and hospitalization in order to precisely analyze the transmission of dengue infection phenomena. The existence and uniqueness are discussed for the model solutions. Basic reproduction number R0 with vaccination is formulated using next generation matrix. Both the local and global stability of disease free and endemic equilibrium points are explored. The parameters are estimated from the actual outbreak of infected cases in Nepal. Sensitivity analysis is then presented graphically with appropriate memory index ψ. The findings of the mathematical and simulation results demonstrate that vaccination is one of the effective strategies that lowers the prevalence of the disease significantly.

Mathematical analysis of fractional-order Caputo’s derivative of coronavirus disease model via Laplace Adomian decomposition method

BackgroundThe world's survival ability has been threatened by the COVID-19 outbreak. The possibility of the virus reemerging in the future should not be disregarded, even if it has been confined to certain areas of the world after wreaking such havoc. This is because it is impossible to prove that the virus has been totally eliminated. This research attempts to investigate the spread and control of the COVID-19 virus in Nigeria using the Caputo fractional order derivative in a proposed model.ResultsWe proposed a competent nine-compartment model of Corona virus infection. It starts by demonstrating that the model is epidemiologically sound in terms of solution existence and uniqueness. The basic reproduction threshold R0 was determined using the next-generation matrix technique. We applied the Laplace-Adomian decomposition method to the fractional-order Caputo's derivative model of the Corona virus disease to produce the approximate solution of the model analytically. The obtained results, in the form of an infinite series, were simulated using the MAPLE 18 package to investigate the effect of fractional order derivative on the dynamics of COVID-19 transmission in the model and shed light on methods of eradication. The graphical interpretations of the simulation process were shown and discussed accordingly.ConclusionsThe study reveals the effect of the Caputo fractional order derivative in the transmission dynamics of the disease. Individual recovery was found to be greatest at an integer order, which represents the full implementation of other factors such as treatment, vaccination, and disease transmission reduction. Hence, we advised that researchers, government officials, and health care workers make use of the findings of this study to provide ways in which disease transmission will be reduced to a minimum to stop the prevalence of COVID-19 by applying the findings of this study.

Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg–de Vries equation

BackgroundExperimentally brought to light by Russell and hypothetically explained by Korteweg–de Vries, the KDV equation has drawn the attention of several mathematicians and physicists because of its extreme substantial structure in describing nonlinear evolution equations governing the propagation of weakly dispersive and nonlinear waves. Due to the prevalent nature and application of solitary waves in nonlinear dynamics, we discuss the soliton solution and application of the fractional-order Korteweg–de Vries (KDV) equation using a new analytical approach named the “Modified initial guess homotopy perturbation.”ResultsWe established the proposed technique by coupling a power series function of arbitrary order with the renown homotopy perturbation method. The convergence of the method is proved using the Banach fixed point theorem. The methodology was demonstrated with a generalized KDV equation, and we applied it to solve linear and nonlinear fractional-order Korteweg–de Vries equations, which are in Caputo sense. The method’s applicability and effectiveness were established as a feasible series of arbitrary orders that accelerate quickly to the exact solution at an integer order and are obtained as solutions. Numerical simulations were conducted to investigate the effect of Caputo fractional-order derivatives in the dispersion and propagation of water waves by varying the order alpha on the [0,1] interval. Comparative analysis of the simulation results, which were presented graphically and discussed, reveals that the degree of freedom of the Caputo fractional-order derivative is vital to controlling the magnitude of environmental hazards associated with water waves when adjusted.ConclusionThe proposed method is recommended for obtaining convergent series solutions to fractional-order partial differential equations. We suggested that applied mathematicians and physicists investigate this work to better understand the impact of the degree of freedom posed by Caputo fractional-order derivatives in wave dispersion and propagation, as physical applications can help divert wave-related environmental hazards.