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Application of Adomian Decomposition Method and Variational Iteration Method for the solution of a fractional (arbitrary) order differential equation

100%

Ray S. S. , Bera R. K.

tom Vol. 14, no 4

1169-1173

In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution to a fractional differential equation with an arbitrary order […]. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions to different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement as that obtained by using the ADM.

Matrix Mittag‑Leffler function in fractional systems and its computation

100%

Matychyn I. , Onyshchenko V.

tom Vol. 66, nr 4

495--500

Matrix Mittag‑Leffler functions play a key role in numerous applications related to systems with fractional dynamics. That is why the methods for computing the matrix Mittag‑Leffler function are so important. The matrix Mittag‑Leffler function is a generalization of matrix exponential function. This implies that some of numerous existing methods for computing the matrix exponential can be adapted for matrix Mittag‑Leffler functions as well. Unfortunately, the technique of scaling and squaring, widely used in computing of the matrix exponential, cannot be applied to matrix Mittag‑Leffler functions, as the latter do not possess the semigroup property. Here we describe a method of computing the matrix Mittag‑Leffler function based on the Jordan canonical form representation. This method is implemented with Matlab code [1].

On the controllability of fractional dynamical systems

100%

Balachandran K. , Kokila J.

2012

tom Vol. 22, no. 3

523-531

This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

Global convergence analysis of impulsive fractional order difference systems

100%

He D. , Xu L.

2018

tom Vol. 66, nr 5

599--604

This paper is designed to deal with the convergence and stability analysis of impulsive Caputo fractional order difference systems. Using the Lyapunov functions, the Z-transforms of Caputo difference operators, and the properties of discrete Mittag-Leffler functions, some effective criteria are derived to guarantee the global convergence and the exponential stability of the addressed systems.

The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption

67%

Povstenko Y. , Klekot J.

tom Vol. 16, nr 2

101--112

The time-fractional heat conduction equation with heat absorption proportional to temperature is considered in the case of central symmetry. The fundamental solutions to the Cauchy problem and to the source problem are obtained using the integral transform technique. The numerical results are presented graphically.