




On the l2 Stability of CrankNicolson Difference Scheme for Time Fractional Heat Equations


Serife R. Bayramoglu Erguner,
Ibrahim Karatay,


Abstract
: 

In this work, the matrix stability of the finite difference scheme based on Crank Nicolson method, for solving timefractional
heat equations, is investigated. An iterative formula is presented for the coefficient matrices of the error equations. Upper bounds for
l2− norms of the coefficient matrices are obtained by a new method based on matrix diagonalization. A detailed numerical analysis,
including tables figures and error comparisons, are given to demonstrate the theoretical results. 






RiemannLiouville Fractional Derivative and Application to Model Chaotic Differential Equations


Kolade M. Owolabi,


Abstract
: 

In this work, stability analysis and numerical treatment of chaotic timefractional differential equations are considered. The
classical system of ordinary differential equations with initial conditions is generalized by replacing the firstorder time derivative with
the RiemannLiouville fractional derivative of order a, for 0 < a ≤ 1. In the numerical experiments, we observed that analysis of
pattern formation in timefractional coupled differential equations at some parameter value is almost similar to a classical process. A
range of chaotic systems with current and recurrent interests which have many applications in biology, physics and engineering are
taken to address any points and queries that may naturally occur. 






BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL AND ANTIPERIODIC CONDITIONS IN A BANACH SPACE


WAFAA BENHAMIDA, JOHN R. GRAEF,SAMIRA HAMANI,


Abstract
: 

The authors study the existence of solutions to a class of fractional differential
equations with antiperiodic and integral boundary conditions involving the
Caputo fractional derivative of order r ∈ (0, 1]. The proof is based on M¨onch’s fixed
point theorem. 






Properties of mixed hyperbolic B–potential


Elina L. Shishkina,


Abstract
: 

n this article a theory of fractional powers of a singular hyperbolic operator
on arbitrary spaces is discussed. Fourier–Hankel transform, semigroup property,
boundedness in proper functional spaces and other properties of the mixed Riesz fractional
integral generated by Bessel operator are obtained. 






On the generalized $(k,\rho)$fractional derivative


D. S. Oliveira,
E. Capelas de Oliveira,


Abstract
: 

We generalize a fractional derivative type, the socalled
$(k,\rho)$fractional derivative. From this generalization, we establish some properties involving this operator. As an application we also show that the
Cauchy problem is equivalent to a Volterra integral equation of second kind. We discuss, from this problem, some particular cases. 






Analytical solution of timefractional NavierStoke equation by natural homotopy perturbation method


Shehu Maitama,


Abstract
: 

An analytical method called the natural homotopy perturbation method for solving timefractional NavierStokes was proposed. The natural homotopy perturbation
method is coupling of a well known technique, homotopy perturbation method (HPM) and the natural transform method (NTM). The proposed analytical method gives a series solutions which converges within few iterations. The efficiency and the high accuracy of the analytical method are clearly illustrated. 






Deformations in ElastoPlastic Media with Memory: the Inverse Problem


Michele Caputo,
Mauro Fabrizio,


Abstract
: 

We consider anelastic media governed by constitutive equations with
memory behavior, which depend on the physical properties of the medium
itself. In this note we use a model of elastoplastic media with two un
specified memory formalisms, which are determined by performing a single
virtual experiment on a sample of the medium. As an application, using a
mathematical memory formally mimicking the CaputoFabrizio fractional
derivative, we show that, when the applied stress is asymptotically vanish
ing, then a shorter memory in the constitutive equation and/or a slower
decay of the applied stress, generate larger asymptotic plastic residual
strain.
In the last part of the paper, we present a nonlinear stressstrain
constitutive equation, which is suitable for describing hysteresis loops with
discontinuity in the first derivative of the cycle. 






The approximate solution of nonlinear fractional optimal control problems by measure theory approach


Ehsan Ziaei,


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 






An Inverse Problem for the Caputo Fractional Derivative by means of the Wavelet Transform


Fabio Marcela,


Abstract
: 

In this article, we build an approximate solution to an Inverse Problem that consist in finding a function whose Caputo
Fractional Derivative is given. We decompose and project the data in appropriate wavelet subspaces and, by a Galerkin scheme, we
calculate the coefficients of the unknown function in the chosen wavelet basis. Based on properties of the operator and of the basis,
the scheme is efficient and the errors introduced by the approximation can be handled and controlled. We illustrate the results with an
example. 






Algorithmic Convergence on Banach Space Valued Functions in Abstract gFractional Calculus


George A. Anastassiou,
Ioannis K. Argyros,


Abstract
: 

The novelty of this paper is the design of suitable algorithms for solving equations on Banach spaces. Some applications of the semilocal convergence are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochnertype.







Iterative Methods and their Applications to Banach Space Valued Functions in Abstract Fractional Calculus


George A. Anastassiou,
Ioannis K. Argyros,


Abstract
: 

Explicit iterative methods have been used extensively to generate a sequence approximating a solution of an equation on a Banach space setting. However, little attention has been given to the study of implicit iterative methods. We present a semilocal convergence analysis for a some general implicit and explicit iterative methods. Some applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochnertype 






New Analytical Solutions and Approximate Solution of The Conformable SpaceTime Fractional SharmaTassoOlver Equation


O. Tasbozan,
Y. C¸ enesiz,
A. Kurt,
O. S. Iyiola,


Abstract
: 

The main purpose of this article is to find the exact and approximate solutions of conformable spacetime fractional SharmaTassoOlver equation using first integral method (FIM) and qhomotopy analysis method (qHAM) respectively. The obtained exact and numerical solutions are compared with each other. Also, the numerical results obtained by qHAM 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 RiemannLiouville derivative


Owolabi Kolade Matthew,


Abstract
: 

Stability analysis and numerical treatment of chaotic fractional differential system in RiemannLiouville 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 Applications of the Fractional Calculus for Some Singular Differential Equations


Resat Yilmazer,


Abstract
: 

Generalized Leibniz rule and some theorems in the calculus of the fractional derivatives and integrals was used to obtain the fractional solutions of the radial Schrödinger equation transformed into a singular differential equation and, these solutions were also exhibited as hypergeometric notations. 






Generalized differential transform method for solving liquidfilm mass transfer equation of fractional order


Elyas Arsanjani Toroqi,


Abstract
: 

Several methods solve linear fractional partial differential equations. In this paper, first, we are presented a fractional model of the liquidfilm mass transfer equation, then the approximate of the generalized differential transform method is compared with the exact solution of the equation in integer orders. Furthermore, the approximates will be obtained in the fractional orders by limiting the intervals of the coefficient and variables. The results show that this method has a high accuracy. 






On the existence and uniqueness of solutions for nonlinear fractional differential equations with fractional boundary conditions


Brahim Tellab,


Abstract
: 

This paper concerns some new existence and uniqueness results obtained by applying classical fixed point theorems for a class of RiemannLiouville fractional differential equations with
fractional boundary conditions. 






Natural Transform Method to Solve Nonhomogeneous Fractional Ordinary Differential Equations


Vinod Gill,
Shikaa Samuel,


Abstract
: 

This paper is committed to solve nonhomogeneous fractional ordinary differential equations both of single and multiple fractional orders where in most cases, the celebrated Mittag Leffeler function is involved in the nonhomogeneous part. A vital result is established and the solution of forced nearly simple harmonic vibration equation is graphically represented amongst others. 




