Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

In this paper, we propose a delayed perturbation of Mittag‐Leffler type matrix function, which is an extension of the classical Mittag‐Leffler type matrix function and delayed Mittag‐Leffler type matrix function. With the help of the delayed perturbation of Mittag‐Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations.     

On the Laplace integral representation of multivariate Mittag‐Leffler functions in anomalous relaxation

In the given paper, a special method of representation of the Mittag‐Leffler functions and their multivariate generalizations in the form of the Laplace integrals is suggested. The method is based on the usage of the generalized multiplication Efros theorem. The possibilities of a new method are demonstrated on derivation of the integral representations for relaxation functions used in the anomalous dielectric relaxation in time domain. Copyright © 2016 John Wiley & Sons, Ltd.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite time stability and relative controllability of Riemann‐Liouville fractional delay differential equations

This paper firstly deals with finite time stability (FTS) of Riemann‐Liouville fractional delay differential equations via giving a series of properties of delayed matrix function of Mittag‐Leffler. We secondly study relative controllability of such type‐controlled system. With the help of the representation of solution, both Gram‐like type matrix and rank criterion are derived, which extend the corresponding results for linear systems.     

Exact and approximate solutions of time-fractional models arising from physics via Shehu transform

In this present investigation, we proposed a reliable and new algorithm for solving time-fractional differential models arising from physics and engineering. This algorithm employs the Shehu transform method, and then nonlinearity term is decomposed. We apply the algorithm to solve many models of practical importance and the outcomes show that the method is efficient, precise, and easy to use. Closed form solutions are obtained in many cases, and exact solutions are obtained in some special cases. Furthermore, solution profiles are presented to show the behavior of the obtained results in other to better understand the effect of the fractional order.     

Global optimization for the sum of generalized polynomial fractional functions

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).     

Analysis of fractional Swift-Hohenberg equation using a novel computational technique

In this paper, the approximated analytical solution for fractional Swift-Hohenberg (S-H) equation is found with the aid of novel technique called q -homotopy analysis transform method (q -HATM) . To ensure the applicability and efficiency of the proposed algorithm, we consider non-linear arbitrary-order S-H equation in presence and absence of dispersive term. The convergence analysis for the projected problem is presented, and the numerical simulations have been conducted to verify the future scheme is reliable and accurate. Further, the effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are presented through plots. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyse the complex problems that arose in science and technology.     

Remarks on some families of fractional-order differential equations

The main purpose of this note is to draw the reader's attention towards some errors and omissions in a recent work involving solutions of some families of fractional-order differential equations, which was published in this Journal (see, for details, [Tomovski ?, Hilfer R, Srivastava HM. Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions. Integral Transforms Spec Funct. 2010;21:797–814]). Several relevant remarks and observations on some other related recent developments on this subject are also presented.     

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.     

Ultimate boundedness of impulsive fractional differential equations

This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results.     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

The calculus of the trigonometric functions

The trigonometric functions entered “analysis” when Isaac Newton derived the power series for the sine in his De Analysi of 1669. On the other hand, no textbook until 1748 dealt with the calculus of these functions. That is, in none of the dozen or so calculus texts written in England and the continent during the first half of the 18th century was there a treatment of the derivative and integral of the sine or cosine or any discussion of the periodicity or addition properties of these functions. This contrasts sharply with what occurred in the case of the exponential and logarithmic functions. We attempt here to explain why the trigonometric functions did not enter calculus until about 1739. In that year, however, Leonhard Euler invented this calculus. He was led to this invention by the need for the trigonometric functions as solutions of linear differential equations. In addition, his discovery of a general method for solving linear differential equations with constant coefficients was influenced by his knowledge that these functions must provide part of that solution.     

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.     

The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs

In this paper, we propose new technique for solving stiff system of ordinary differential equations. This algorithm is based on Laplace transform and homotopy perturbation methods. The new technique is applied to solving two mathematical models of stiff problem. We show that the present approach is relatively easy, efficient and highly accurate.     

Exact solutions and finite time stability for higher fractional-order differential equations with pure delay

We obtain the exact solutions to the higher fractional-order nonhomogeneous delayed differential equations with Caputo-type fractional derivative by using a set of newly defined generalized delayed Mittag-Leffler matrix functions. The Laplace transform and inductive construction are taken up as the major solving approaches. Thereafter, we consider some special cases and prove that the new exact solutions are suitable for the delayed differential equations with arbitrary order . Additionally, we propose some criteria on the finite time stability of the higher fractional-order delay differential equations. Finally, an illustrative example is presented to test the correctness of the theoretical results.     

Certain fractional calculus and integral transform results of incomplete ℵ-functions with applications

In this paper, we study certain interesting and useful properties of incomplete $\aleph$-functions. The incomplete $\aleph$-function is an extension of the $\aleph$-function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete $\aleph$-functions and point out several special cases. Finally, we give the applications of incomplete $\aleph$-functions in detecting glucose supply in human blood.     

有限分形介质中分数阶反应扩散方程及其解析解

建立了有限分形介质中具有吸附效应的分数阶反应扩散积分方程.利用Lap lace变换、广义有限H ankel变换及其相应的逆变换得到了以M ittag-Leffler函数为主要形式的解析解,并研究了解的渐近性态.     

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie‐modified Riemann–Liouville derivative

In this article, the fractional variational iteration method is employed for computing the approximate analytical solutions of degenerate parabolic equations with fractional time derivative. The time‐fractional derivatives are described by the use of a new approach, the so‐called Jumarie modified Riemann–Liouville derivative, instead in the sense of Caputo. The approximate solutions of our model problem are calculated in the form of convergent series with easily computable components. Moreover, the numerical solution is compared with the exact solution and the quantitative estimate of accuracy is obtained. The results of the study reveal that the proposed method with modified fractional Riemann–Liouville derivatives is efficient, accurate, and convenient for solving the fractional partial differential equations in multi‐dimensional spaces without using any linearization, perturbation or restrictive assumptions. Copyright © 2011 John Wiley & Sons, Ltd.