Solving fractional optimal control problems by new Bernoulli wavelets operational matrices

Abstract

SummaryIn this article, a new numerical method based on Bernoulli wavelet basis has been applied to give the approximate solution of the fractional optimal control problems. The new operational matrices of multiplication and fractional integration are constructed. The proposed method is applied to reduce the problem to the solution of a system of algebraic equations. The fractional derivative is considered in the Caputo sense. Convergence of the algorithm is proved and some results concerning the error analysis are obtained. Approximate solutions are given and in the cases when we have an exact solution, a comparison with the exact solution is presented to demonstrate the validity and applicability of the proposed method. In addition, we compare the obtained results with the results of other methods. Comparison shows the more accuracy of presented technique in comparison to other published methods.

Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis

“…It is obvious Qm˜⊆Qtruem˜+1, this implies ξm˜≥ξtruem˜+1. Thus, it shows the convergence of ξ because ξm˜ is a nonincreasing and bound sequence (Barikbin and Keshavarz, 2020).…”

“…Consider the FOCP as followssubject toand x (0) = 1. The exact solution of the considered problem in ν = 1 is (Barikbin and Keshavarz, 2020; Hager, 1976)the exact value of functional is J=(exp(2)sinh(2))/(1+exp(2))2≃0.380797078. We implement the present framework to solve the problem with M = 4 and various values of k .…”

“…The absolute errors of the state and control variables x ( t ) and u ( t ) for ν = 1 with M = 4 and k = 2, 3 are shown in Figure 6. Also, Table 1 shows the approximate functional using the proposed method, Bernoulli wavelets (Barikbin and Keshavarz, 2020), Boubaker polynomials (Rabiei et al, 2017), and Chebyshev finite difference method (El-Kady, 2003).…”

“…In Agrawal et al (2010), the authors presented a method to obtain approximated solution of FOCPs in the Riemann–Liouville derivative. Two numerical methods were proposed in Barikbin and Keshavarz (2020) and Keshavarz et al (2016) for solving this class of problems in the Caputo sense. A method based upon Boubaker wavelets was proposed for finding numerical solution of FOCPs in Rabiei and Ordokhani (2020).…”

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

“…The key feature of the approach is applying the Ritz-Galerkin technique [66][67][68][69][70][71][72][73][74] along with utilizing the satisfier function to fulfill the initial and Dirichlet boundary conditions. As a consequence, only a low number of bases is required to find accurate approximation with reliably less computations.…”