Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 Continuous Family of Solutions for Fractional Integro-Differential Inclusions of Caputo-Katugampola Type Abstract : A continuous family of solutions for a fractional integro-differential inclusion involving Caputo-Katugampola fractional derivative is obtained.

 Mixed Conformable and Iterated fractional Approximation by Choquet integrals Abstract : Here we study the quantitative mixed conformable and iterated frac- tional approximation of positive sublinear operators to the unit opera- tor. These are given a precise Choquet integral interpretation. Initially we start with the research of the mixed conformable and iterated frac- tional rate of the convergence of the well-known Bernstein-Kantorovich- Choquet and Bernstein-Durrweyer-Choquet polynomial Choquet-integral operators. Then we study the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler (1986) [13]. We …nish with the approximation by the very general direct Choquet- integral form positive sublinear operators. All mixed conformable and iterated fractional approximations are given via inequalities involving the modulus of continuity of the approximated function mixed conformable and iterated fractional order derivative.

 Caputo and Canavati fractional Approximation by Choquet integrals Abstract : Here we consider the quantitative Caputo and Canavati fractional ap- proximation of positive sublinear operators to the unit operator. These are given a precise Choquet integral interpretation. Initially we start with the study of the fractional rate of the convergence of the well-known Bernstein-Kantorovich-Choquet and Bernstein-Durrweyer-Choquet poly- nomial Choquet-integral operators. Then we study the very general comonotonic positive sublinear operators based on the representation theorem of Schmei- dler (1986) [17]. We …nish with the approximation by the very general direct Choquet-integral form positive sublinear operators. All fractional approximations are given via inequalities involving the modulus of conti- nuity of the approximated function fractional order derivative.

 Numerical simulation for system of time-fractional linear and nonlinear differential equations Abstract : This paper is concerned with \textit{q}-homotopy analysis transform technique to examine system of differential equations of fractional order. The proposed technique describes the convergence range at large domain, by appropriate selection of initial approximation, auxiliary parameter and asymptotic parameter n ($n\geq1$). The proposed technique provides infinitely many more options for solution series and converge rapidly compared to homotopy analysis method(HAM), homotopy perturbation transform algorithm(HPTA) in same term iterations. A comparative study of suggested scheme with exact, HAM and HPTA have been done and Maple package is used to enhance the power and efficiency of proposed technique.

 Extension of incomplete gamma, beta and hypergeometric functions Abstract : Recently, some generalizations of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been introduced in Ozergin. In this paper we introduce generalizations of incomplete gamma, beta, Gauss hypergeometric and confluent hypergeometric functions. Some integral representations, Mellin transforms, transformation formulas, differentiation and difference formulas are obtained for these functions.

 Experimental Verification of Fractional Modelling of the Viscoelastic Response in Polymer Biomaterials Abstract : Viscoelastic mechanical systems are modelled with the help of Leibniz fractional (L-Fractional) derivative. Since this derivative has important physical and mathematical meaning, it would be interesting to compare the theoretical with experimental data. The relaxation behaviour of the Zener viscoelastic model is presented, and compared with experimental data. The experimental results of the viscoelastic relaxation behaviour in a polymer mesh used for the surgical treatment of female urinary incontinence were used in order to check the applicability of fractional modelling in these systems. Data from relaxation experiments were used in combination with theoretical analysis to prove the Zener model fractional analysis concept.

 A numerical method for solving a class of first order fractional nonlinear Volterra integro-differential type of singularly perturbed problems Abstract : In this paper, we study a class of first order fractional nonlinear Volterra integro-differential type of singularly perturbed problems with fractional order. The method consists of two steps: The first step is to generate the reduced problem when ε=0. We solve it directly if it is possible or to use the finite difference method with Trapezoidal rule when it is not possible to solve it explicitly. The second step is fractional Volterra integro-differntial problem which we solve it by the reproducing kernel method. We prove the stably and the uniqueness of the solution for this class of problems. Numerical results are presented to show the efficiency of the proposed method.

 Numerical Treatment for Solving the Fractional Two Groups Influenza Model Abstract : In this article, a general model for Influenza of two groups is presented as a fractional order model. The fractional derivatives for this model which consist of eight differential equations are defined in the sense of Caputo definition. To obtain an efficient numerical method, the fraction order derivatives are approximated by the shifted Jacobi polynomials. The proposed scheme reduces the solution of the main problem to the solution of a system of nonlinear algebraic equations. Comparative studies between the proposed method and both the fourth-order Runge-Kutta method and the generalized Euler method are done.

 On the local M-derivative Abstract : We introduce a new local derivative that generalizes the so-called alternative $"fractional"$ derivative recently proposed. We denote this new differential operator by $\mathscr{D}_{M}^{\alpha,\beta}(\cdot)$, where the parameter $\alpha$, associated with the order, is such that $0<\alpha<1$, $\beta>0$ and $M$ is used to denote that the function to be derived involves a Mittag-Leffler function with one parameter. This new derivative satisfies some properties of integer-order calculus, e.g. linearity, product rule, quotient rule, function composition and the chain rule. Besides as in the case of the Caputo derivative, the derivative of a constant is zero. Because Mittag-Leffler function is a natural generalization of the exponential function, we can extend some of the classical results, namely: Rolles theorem, the mean value theorem and its extension. We present the corresponding $M$-integral from which, as a natural consequence, new results emerge which can be interpreted as applications. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts. Finally, we present an application involving linear differential equations by means of local $M$-derivative with some graphs.

 Hilfer-Hadamard Fractional Di erential Equations and Inclusions Under Weak Topologies Abstract : In this article, by applying some Monchs xed point theorems associated with the technique of measure of weak noncompactness, we prove some results concerning the existence of weak solutions for some Hilfer-Hadamard fractional di erential equations and inclusions.

 Solutions for some conformable fractional differential equations Abstract : This paper deals with the analytic candidate solutions for conformable fractional differential equations. We give candidate solutions for fractional differential equations of order $\alpha$ and $2\alpha$. Integration will be the key of this paper. Many examples are given to illustrate our main results of this paper.

 Dynamical analysis of a fractional-order Rosenzweig-MacArthur model with stage structure incorporating a prey refuge Abstract : In this paper, a fractional order prey-predator model with stage structure incorporating a prey refuge is constructed and analyzed. The predators are divided into immature and mature predators using Holling type II functional response. The existence, uniqueness, non-negativity and boundedness of the model as well as the local and global stability of the equilibrium points are investigated. Sufficient conditions for the stability and Hopf bifurcation for the fractional order model are obtained. The impact of fractional order, a prey refuge and conversion coefficient on the stability of the fractional order system are also studied both theoretically and numerically.

 System Identification with Fractional-order Models: A Comparative Study with Different Model Structures Abstract : The use of fractional order models are growing in the research field of modeling. However, there is no attempt to compare different fractional order models. In this paper, a comparison of different fractional model structures is presented with a simulation of various systems. The various model structures covers classical model, classical model with zero, commensurate, and non-commensurate fractional model. The results of fractional model structures are also compared with an integer order model structure. Simulation results shows that non-commensurate fractional model is performing better than the other fractional and integer model structures.

 A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function Abstract : While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac delta function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemanns complimentary function introduced during his research on fractional derivatives.

 New Analytical Solutions and Approximate Solution of The Conformable Space-Time Fractional Sharma-Tasso-Olver Equation Abstract : The main purpose of this article is to find the exact and approximate solutions of conformable space-time fractional Sharma-Tasso-Olver equation using first integral method (FIM) and q-homotopy analysis method (q-HAM) respectively. The obtained exact and numerical solutions are compared with each other. Also, the numerical results obtained by q-HAM are compatible with the exact solutions obtained by FIM. Hence, it is clearly seen that these techniques are powerful and efficient in finding approximate and exact solutions for nonlinear conformable fractional PDEs.

 Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative Abstract : Stability analysis and numerical treatment of chaotic fractional differential system in Riemann-Liouville sense are considered in this paper. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise.

 On the existence and uniqueness of solutions for nonlinear fractional differential equations with fractional boundary conditions Abstract : This paper concerns some new existence and uniqueness results obtained by applying classical fixed point theorems for a class of Riemann-Liouville fractional differential equations with fractional boundary conditions.