A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

The sine and cosine diffusive representations for the Caputo fractional derivative

As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end

A matrix approach to multi-term fractional differential equations using two new diffusive representations for the Caputo fractional derivative

In the last decade, there has been a surge of interest in application of fractional calculus in various areas such as, mathematics, physics, engineering, mechanics and etc. So, numerical methods have rapidly been developed to handle problems containing fractional derivatives (or integrals). Due to the fact that all the operators which appear in fractional calculus are non-local, so, the classical linear multi-step methods have some difficulties from the (time/space) computational complexity point of view. Recently, two new non-classical methods or diffusive based methods have been proposed by the authors to approximate the Caputo fractional derivatives. Here, the main aim of this paper is to use these methods to solve linear multi-term fractional differential equations numerically. To reach our aim, an efficient matrix approach has been provided to solve some well-known multi-term fractional differential equations

Numerical solution for fractional variational problems using the Jacobi polynomials

We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By some examples, we show the convergence of such procedure, comparing the exact solution with numerical approximations

The sine and cosine diffusive representations for Caputo fractional derivative: A matrix approach to multi-term fractional differential equations

Abstract In the last decade, there has been a surge of interest in application of fractional calculus in various areas such as, mathematics, physics, engineering, mechanics and etc. So, numerical methods have rapidly been developed to handle problems containing fractional derivatives (or integrals). Due to the fact that all the operators which appear in fractional calculus are non-local, so, the classical linear multi-step methods have some diffculties from the point of view of (time/space) computational complexity. Recently, two newnon-classical methods or diffusive based methods have been proposed by the authors to approximate the Caputo fractional derivatives. Here, the main aim of this paper is to use these methods to solve linear multi-term fractional differential equations numerically. To reach our aim, an effcient matrix approach hasbeen provided to solve some well-known multi-term fractional differential equations. Mathematics Subject Classifications (2000): 26A33; 65D30; 65D25; 65D32.</jats:p