Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 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.

 Fractional Calculus of Wright Function With Raizada Polynomial Abstract : The aim of this paper is to established certain generalized fractional integration and di erentiation formula to the product of Wright func- tion and Raizada polynomial. The results are obtained in compact form containing the Riemann-Liouville , Erdelyi-Kober and Saigo operators of fractional calculus.

 Canavati Fractional Approximation by Max-product Operators Abstract : Here we study the approximation of functions by sublinear positive operators with applications to a large variety of Max- Product operators under Canavati fractional differentiability. Our approach is based on our general fractional results about positive sublinear operators. We derive Jackson type inequalities under simple initial conditions. So our way is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of Canavati fractional derivative of the function under approximation.

 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.

 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.

 Local Fractional Natural Homotopy Perturbation Method for Solving Partial Differential Equations with Local Fractional Derivative Abstract : A new analytical method called the Local Fractional Natural Homotopy Perturbation Method (LFNHPM) for solving partial differential equations with local fractional derivative is introduced. The proposed analytical method is a combination of the homotopy perturbation method (HPM) and the natural transform (NTM). In this analytical method, the fractional derivative operators are computed in local fractional sense. Some applications are given to illustrate the simplicity, efficiency, and high accuracy of the Local Fractional Natural Homotopy Perturbation Method.

 A New Approach for Solving Fractional Optimal Control Problems Using Shifted Ultraspherical Polynomials Abstract : Through this article, we introduce a numerical technique for solving one and two dimensional fractional optimal control problems (FOCPs). The proposed technique based on using the operational matrix (OM) of the Riemmaan– Liouville (RL) fractional integral of the shifted Gegenbauer polynomials (SGPs) to transform the FOCP into an equivalent variational problem. By using the Gegenbauer- Gauss quadrature method (GGQM) with the Rayleigh- Ritz method (RRM) the resultant variational problem is reduced to a system of algebraic equations (AEs) which is easily to solve. Illustrative problems included one and two dimensional FOCPs are provided to illustrate the powerful and validity of our method.

 The approximate solution of nonlinear fractional optimal control problems by measure theory approach Abstract : In this paper a novel strategy in finding an approximate solution for nonlinear fractional optimal control is proposed. The measure theory is used to find the solution. There are different definition for fractional derivative and integral, in this work new definition which is named conformable fractional calculus is used. Analytically convergence of the proposed method is proved, finally

 Existence of smooth solutions of multi-term Caputo-type fractional differential equations Abstract : This paper deals with the initial value problem for the multi-term fractional differential equation. The fractional derivative is defined in the Caputo sense. Firstly the initial value problem is transformed into a equivalent Volterra-type integral equation under appropriate assumptions. Then new existence results for smooth solutions are established by using the Schauder fixed point theorem.

 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.

 Direct and inverse problems for a Samarskii-Ionkin type problem for a two dimensional fractional parabolic equation Abstract : In this work, we investigate a direct and an inverse nonlocal problems of Samarskii-Ionkin type for a two dimensional fractional parabolic equation. Existence and uniqueness of solutions of the considered problems is presented. The proof of the obtained results relies on the Fourier method of the separation of variables and bi-orthogonal series expansion of the solution.

 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.