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In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods. [ABSTRACT FROM AUTHOR]

The creation of an explicit finite difference scheme with the express purpose of resolving initial boundary value issues with linear and semi-linear variable-order temporal fractional properties is presented in this study. The rationale behind the utilization of the Caputo derivative in this scheme stems from its known importance in fractional calculus, an area of study that has attracted significant interest in the mathematical sciences and physics. Because of its special capacity to accurately represent physical memory and inheritance, the Caputo derivative is a relevant and appropriate option for representing the fractional features present in the issues this study attempts to address. Moreover, a detailed Fourier analysis of the explicit finite difference scheme's stability is shown, demonstrating its conditional stability. Finally, certain numerical example solutions are reviewed and MATLAB-based graphic presentations are made. [ABSTRACT FROM AUTHOR]

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on R , we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed: 1 ∇ h 2 u (t) + A C ∇ h ν u (t) + B u (t) = f (t) , t > 0 , where 0 < ν < 1 or 1 < ν < 2 , subject to u (0) = a and ∇ h u (0) = b , with a and b being real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters. [ABSTRACT FROM AUTHOR]

Nonlocal elasticity assumes integral stress constitutive equation, takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. In this paper, we propose the nonlocal elasticity theory with the kernel (the weight function in the integral stress constitutive equation) which is the Green function of the Cauchy problem for the fractional diffusion equation and is expressed in terms of the Mainardi function and Wright function being the generalizations of the exponential function. The solutions for the straight screw and edge dislocations in an infinite solid are obtained in the framework of the proposed theory. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that a time fractional reaction diffusion system with reaction terms of the Brusselator type admits a global solution by using the feedback method of F. Rothe [20]. Furthermore, some results on the large time behavior of the solutions are obtained. We give a positive answer to Problem 6 of the valuable paper of Gal and Warma [6]. [ABSTRACT FROM AUTHOR]

We examine the transition region between two solids which state differs from the state of contacting media. Small thickness of the intermediate region allows us to reduce a three-dimensional problem to a two-dimensional one for a median surface endowed with equivalent physical properties. In the present paper, we consider the generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction equation with the Caputo derivative and solve the problem for a composite medium consisting of two semi-infinite regions. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

We use the Laplace transform method to solve certain families of fractional order differential equations. Fractional derivatives that appear in these equations are defined in the sense of Caputo fractional derivative or the Riemann-Liouville fractional derivative. We first state and prove our main results regarding the solutions of some families of fractional order differential equations, and then give examples to illustrate these results. In particular, we give the exact solutions for the vibration equation with fractional damping and the Bagley-Torvik equation. [ABSTRACT FROM AUTHOR]

This paper is dedicated to several original (weighted) Hölder continuity results for Riemann-Liouville fractional integrals of weighted integrable functions. As an application, we prove a new weighted continuity result for solutions to nonlinear Riemann-Liouville fractional Cauchy problems with Carathéodory dynamics. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory. [ABSTRACT FROM AUTHOR]

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

This work studies the approximate controllability of a class of impulsive neutral stochastic differential equations with infinite delay and Poisson jumps involving generalized Caputo fractional derivative under the condition that the corresponding linear system is approximately controllable. Utilizing the fixed point theory and sectorial operator theory, the existence of the mild solution of the impulsive neutral stochastic equation is established imposing weaker regularity on nonlinear terms. A set of sufficient conditions establishing controllability results is derived with the help of stochastic analysis and fractional calculus. Finally, an example is provided to illustrate the obtained abstract result. [ABSTRACT FROM AUTHOR]

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

For a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods. [ABSTRACT FROM AUTHOR]

In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation. [ABSTRACT FROM AUTHOR]

The generalized fractional integral operator and the Caputo type generalized fractional derivative operator are defined which contain the Riemann-Liouville integral operator, the Hadamard fractional integral operator, the Caputo fractional derivative operator and the Caputo type Hadamard fractional derivative operator as special cases. General solutions (the explicit solutions) of the impulsive Caputo type generalized fractional differential equations are given. Applying our results, existence results of solutions of boundary value problems for an impulsive fractional differential equations involved with the Caputo type generalized fractional derivatives are established. Examples and some remarks on recent published papers are presented to illustrate the main theorems. [ABSTRACT FROM AUTHOR]

We have applied the new approach of homotopic perturbation method ( NHPM) for Schrödinger equation featuring time-fractional derivative. A combination of NHPM, Laplace transform and multiple fractional power series form has been used the first time to present analytical solution. Then, in order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica. [ABSTRACT FROM AUTHOR]

In this article, we investigate some uniqueness and Ulam's type stability concepts for the Darboux problem of partial functional differential equations with noninstantaneous impulses and delay in Banach spaces. The main techniques rely on fractional calculus, integral equations and inequalities. Two examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the classical control theory to a fractional differential system in a bounded domain. The fractional optimal control problem (FOCP) for differential system with time delay is considered. The fractional time derivative is considered in a Riemann-Liouville sense. We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a FOCP is considered as a function of both state and control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of a right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in detail. [ABSTRACT FROM AUTHOR]

In this paper, a wavelet numerical method for solving nonlinear Volterra integro-differential equations of fractional order is presented. The method is based upon Euler wavelet approximations. The Euler wavelet is first presented and an operational matrix of fractional-order integration is derived. By using the operational matrix, the nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which is solved through known numerical algorithms. Also, various types of solutions, with smooth, non-smooth, and even singular behavior have been considered. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to develop and analyze a new high-order operational Tau method based on the Chebyshev polynomials as basis functions for obtaining the numerical solution of Bagley-Torvik equation which has a important role in the fractional calculus. It is shown that some derivatives of the solutions of these equations have a singularity at origin. To overcome this drawback we first change the original equation into a new equation with a better regularity properties by applying a regularization process and thereby the operational Chebyshev Tau method can be applied conveniently. Our proposed method has two main advantages. First, the algebraic form of the Tau discretization of the problem has an upper triangular structure which can be solved by forward substitution method. Second, Tau approximation of the problem converges to the exact ones with a highly rate of convergence under a more general regularity assumptions on the input data in spite of the singularity behavior of the exact solution. Numerical results are presented which confirm the theoretical results obtained and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional-order Rayleigh oscillator model, which involves a cubic damping term described by fractional derivatives. The presence of such fractional damping term makes the analysis more difficult. A two-scale expansion method is employed for asymptotic solutions of the fractional-order Rayleigh oscillator. Then, an example is provided to compare the asymptotic solutions with the numerical solutions. The numerical results demonstrate the validity and applicability of the proposed method to solve fractional differential equations with high order fractional terms. Furthermore, an electronic circuit is designed to realize the fractional-order Rayleigh oscillator. [ABSTRACT FROM AUTHOR]

This paper mainly concerns with the existence of a mild solution for impulsive neutral integro-differential inclusions with nonlocal conditions in a separable Hilbert space. Utilizing fixed point theorem for multi-valued operators due to Dhage, we establish the existence result with resolvent operator and η-norm. An illustrative example is provided to show the effectiveness of the established results. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of a mild solution of a fractional order nonlocal differential inclusion with impulsive condition in a Banach space E. We obtain the sufficient condition for the existence of the mild solution by using a fixed point theorem for multi-valued operators due to Dhage and resolvent semigroup theory with approximate techniques. [ABSTRACT FROM AUTHOR]

We present an analysis of projectile motion in view of fractional calculus. We obtain the solution for the problem using the Riemann-Liouville derivative, and then we compute some features of projectile motion in the framework of Riemann-Liouville fractional calculus. We compare the solutions using Caputo derivatives and Riemann-Liouville derivatives. [ABSTRACT FROM AUTHOR]

In this work we report the development of an implicit finite difference numerical method for the one space dimension time-fractional advection-diffusion equation, on a bounded domain, to model the transient electrical current of the time of flight experiment of disordered (e.g. organic) semiconductors. Some numerical experiments and simulation of experimental data are carried out showing that the presented model describes accurately the transient electrical current. [ABSTRACT FROM AUTHOR]

In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R, L and C an auxiliary parameter γ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order of the fractional differential equation. [ABSTRACT FROM AUTHOR]

This paper provides an analysis in the time and frequency domain of an RC electrical circuit described by a fractional differential equation of the order 0 < α≤ 1. We use the Laplace transform of the fractional derivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times, while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes in low and high frequencies. All these quantities depend on the order of differential equation. [ABSTRACT FROM AUTHOR]

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$, where D is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay. [ABSTRACT FROM AUTHOR]

The objectives of this work are (i) to verify experimentally the theoretical claim that neither Riemann-Liouville nor Caputo fractional derivative can be used to predict the time response of fractional order systems without proper correction for the system's past history in terms of an initialization function and (ii) to study quantitatively how the error incurred due to ignoring initialization depends on the nature of the past history and the system parameters. The entire analysis is restricted to a special class of single input single output linear time invariant fractional order system which can be realized by a simple electrical circuit consisting of a resistance and a fractance in series. This work involves two different realizations of fractances or constant phase elements whose characteristic parameters are first determined based on their respective impedance frequency responses and then used to simulate the time responses of the circuit with the input same as the one used for experimentation using a numerical method for two cases: (i) taking past history into account and (ii) without taking past history into account. Thereafter, an integral square error criterion is presented and variation of the same is studied with respect to system parameters and nature of the history function to have a relative idea of how much partial past history in the absence of a complete one should suffice in practical applications. [ABSTRACT FROM AUTHOR]

We propose a new mathematical model for the simulation of the dynamics of a dengue fever outbreak. Our model differs from the classical model in that it involves nonlinear differential equations of fractional, not integer, order. Using statistics from the 2009 outbreak of the disease in the Cape Verde islands, we demonstrate that our model is capable of providing numerical results that agree very well with the real data. [ABSTRACT FROM AUTHOR]

In the present paper the effect of delay on chaos in fractional-order Chen system is investigated. It is observed that inclusion of delay changes chaotic behaviour to limit cycles or stable systems. [ABSTRACT FROM AUTHOR]

In this paper, we present a framework to obtain the solutions to the fractional vibration equation by the homotopy perturbation method. The fractional derivative is described in the Caputo sense. Our method performs extremely well in terms of efficiency and simplicity. Numerical results are presented graphically showing the complete reliability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the complex problem of how to simulate particle contacts, taking into account the cohesion effect. In accordance with the molecular dynamics models, we propose a novel expression for the repulsive force which controls dynamically the transfer and dissipation of energy in granular media. This expression is formulated under fractional calculus, where a fractional derivative accumulates the whole history of the virtual overlap over time in weighted form. We then discuss and illustrate the basic properties of the repulsive force in a normal direction to the contacting surfaces. This approach allows us to perform simulations of arbitrary multiparticle contacts as well as granular cohesion dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we derive some sufficient conditions which ensure the existence and uniqueness of a solution for a class of nonlinear three point boundary value problems of fractional order implicit differential equations (FOIDEs) with some boundary and impulsive conditions. Also we investigate various types of Hyers-Ulam stability (HUS) for our concerned problem. Using classical fixed point theory and nonlinear functional analysis, we obtain the required conditions. In the last section we give an example to show the applicability of our obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we derive the compositions of the fractional derivatives with the Shukla function, a four-parameter Mittag-Leffler function. We investigate and compare the difference between the Riemann-Liouville and Caputo derivatives of the generalized Mittag-Leffler functions and obtain the reason causing the difference and expand the fractional derivatives of the generalized Mittag-Leffler functions. Two illustrative examples and the related numerical results are provided to demonstrate the validity. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method called the inverse fractional natural transform method (IFNTM). We present theoretical results and apply them to obtain approximate solutions of linear fractional ordinary differential equations (LFODEs) and partial differential equations (LFPDEs). The fractional derivatives are described in the Caputo sense. The algorithm described in this study is expected to be further employed to solve similar linear problems in fractional calculus. [ABSTRACT FROM AUTHOR]