A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

Abstract

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann-Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.Keywords: fractional integration operational matrix of Chebyshev wavelets; Chebyshev wavelets method; fractional time-delay optimal control; multifractional delay differential equation; quadratically constrained linear-quadratic optimal control; piecewise delay

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

“…The Riemann-Liouville fractional integral operator of order α denoted by I α of a function f (t) is defined by [11]…”

“…Also, some of the methods are only applicable to a specific class of nonlinear optimal control problems, for example see [10]. We proposed a simple method in [11] based on an exact Riemann-Liouville fractional integration operational matrix of Chebyshev wavelets to obtain the optimal control of fractional linear quadratic systems from solving quadratic optimization problems and without doing any work, the value of optimal cost is reached as a default output of the quadprog solver. Whereas some of IMAN MALMIR 859 real-world systems are modeled by nonlinear differential equations, for example a model of machine tool vibrations in the turning of metals [12], and differential equations of fractional orders are valuable tools in modeling of some phenomena in various fields of engineering [13,14,15], now the question is: Whether the method can be applied to fractional nonlinear systems?…”

“…Whereas some of IMAN MALMIR 859 real-world systems are modeled by nonlinear differential equations, for example a model of machine tool vibrations in the turning of metals [12], and differential equations of fractional orders are valuable tools in modeling of some phenomena in various fields of engineering [13,14,15], now the question is: Whether the method can be applied to fractional nonlinear systems? Of course, we implicitly implemented the method on a simple nonlinear example in [11]. In this research, we go into greater detail and illustrate how by a general useful wavelet-based method we can solve such problems.…”

Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials

“…Definition 1. The Caputo fractional derivative operator D α(x,t) of order α(x, t) is defined as [22,26]…”

“…Baleanu et al [21] solved the Helmholtz equation based on local fractional derivative operators. Malmir [22] applied a new fractional integration operational matrix of Chebyshev wavelets to solve fractional delay equations. A Bernstein polynomial numerical method for solving a class of variable fractional order linear cable equations [23] and variable order time fractional diffusion equations [24] was proposed.…”