Numerical investigation of distributed‐order fractional optimal control problems via Bernstein wavelets

Abstract

SummaryThe aim of this article is to investigate an efficient computational method for solving distributed‐order fractional optimal control problems. In the proposed method, a new Riemann‐Liouville fractional integral operator for the Bernstein wavelet is given. This approach is based on a combination of the Bernstein wavelets basis, fractional integral operator, Gauss‐Legendre numerical integration, and Newton's method for solving obtained system. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed method. The error analysis of the proposed method is carried out. Examples reveal the applicability of the proposed technique.

Numerical solutions for distributed-order fractional optimal control problems by using Müntz–Legendre wavelets

“…In case β=0.5, by using a similar argument we get C=false[π/2,0,0false]T which yields the exact solutions xfalse(tfalse)=t1/2 and ufalse(tfalse)=t1/2+1/2π. These exact solutions were not obtained in [9] and [27].…”

“…Several examples are given to show the advantage of our method in comparison with Legendre collocation method in [8], Bernstein wavelet method in [9] and several wavelets techniques in [27].…”

“…Consider the DO-FOCP [9,27]: min1emJfalse(x,ufalse)=12∫01false[false(xfalse(tfalse)−tβfalse)2+false(ufalse(tfalse)−tβ−Γfalse(β+1false)false)2false]dt, with ∫01ρfalse(μfalse)scriptDμxfalse(tfalse) dμ=−xfalse(tfalse)+ufalse(tfalse),1emt∈false[0,1false], and xfalse(0false)=1. This example has been solved in [27], with the distribution functions ρfalse(μfalse)=δfalse(μ−βfalse) for β=1 and 0.5, by using several wavelet methods such as: Chebyshev (CWM), cosine and sine (CASWM), Laguerre (LaWM), Legendre (LWM) and variational iteration method (VIM). This problem has also been solved in [9] by using the following distribution functions: right left right left right left right left right left right left3pt0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0emtrueCase 1: ρ(μ...…”

“…If the FOCPs contain distributed-order fractional derivatives, they are called distributed-order FOCPs (DO-FOCPs). Only a few numerical schemes were proposed in the literature to solve DO-FOCPs, such as Legendre collocation method and Bernstein wavelets method in [8,9], respectively.…”

“…Wavelets have been a strong tool in function approximation for solving problems in many areas. Notable wavelets used include CAS [10], Legendre [11], Bernstein [9], Chebyshev [12] and Haar wavelets [13]. In fractional calculus problems, the operational matrices scriptPβ of the wavelet Ψfalse(tfalse) have been used for approximating scriptIβΨfalse(tfalse) as scriptIβΨfalse(tfalse)≅scriptPβΨfalse(tfalse), where scriptIβ is the Riemann–Liouville fractional integral operator (RLFIO) of order β.…”