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Progress in Fractional Differentiation and Applications
An International Journal



On the l2 Stability of Crank-Nicolson 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 time-fractional 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.


Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations

Kolade M. Owolabi,
Abstract :
In this work, stability analysis and numerical treatment of chaotic time-fractional differential equations are considered. The classical system of ordinary differential equations with initial conditions is generalized by replacing the first-order time derivative with the Riemann-Liouville fractional derivative of order a, for 0 < a ≤ 1. In the numerical experiments, we observed that analysis of pattern formation in time-fractional 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.



Abstract :
The authors study the existence of solutions to a class of fractional differential equations with anti-periodic 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 so-called $(k,\rho)-$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 time-fractional Navier-Stoke equation by natural homotopy perturbation method

Shehu Maitama,
Abstract :
An analytical method called the natural homotopy perturbation method for solving time-fractional Navier-Stokes 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 Elasto-Plastic 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 elasto-plastic 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 Caputo-Fabrizio 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 non-linear stress-strain 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 g-Fractional 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 semi-local convergence are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.


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 semi-local 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 Bochner-type


New Analytical Solutions and Approximate Solution of The Conformable Space-Time Fractional Sharma-Tasso-Olver 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 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

Owolabi Kolade Matthew,
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 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 liquid-film 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 liquid-film 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 Riemann-Liouville 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.

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