Summary In this paper, a numerical method for solving time‐varying delay equations and optimal control problems with time‐varying delay systems is discussed. This method is based upon Fibonacci wavelets and Petrov‐Galerkin method. To solve these problems, first, the Fibonacci wavelets are presented. With the aid of operational matrices of integration and delay for Fibonacci wavelets and using Petrov‐Galerkin method and Newton's iterative method, we solve two classes of time‐varying delay problems, numerically. The approximate solutions achieved by this method satisfy all the initial conditions. In addition, an estimation of the error is given. Numerical results are included to demonstrate the accuracy and applicability of the present technique.

The main purpose of this work is to provide an efficient method for solving delay fractional optimal control problems (DFOCPs). Our method is based on fractional-order Lagrange polynomials (FLPs) and the collocation method. The FLPs are used to achieve a new operational matrix of fractional derivative. Also, we present a delay operational matrix of FLPs. These operational matrices are driven without considering the nodes of Lagrange polynomials. The operational matrices and collocation method are applied to a constrained extremum in order to minimize the performance index. Then, the problem reduces to the solution of a system of algebraic equations. Convergence of the algorithm and approximation of FLPs are proposed. Furthermore, the upper bound of the error for the operational matrix of fractional derivatives is obtained. Numerical tests for demonstrating the efficiency and effectiveness of the method are included. Moreover, the method is used for numerical solution of a mathematical model of chemotherapy in breast cancer.

This study presents a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints. The proposed procedure is based upon Fibonacci wavelets. The fractional derivative is described in the Caputo sense. The Riemann–Liouville operational matrix for Fibonacci wavelets is obtained. Then, we use this operational matrix and the Galerkin method to reduce the given problem into a system of algebraic equations. We discuss the convergence of the algorithm. Several numerical examples are included to observe the validity, effectiveness, and accuracy of the suggested scheme. Moreover, fractional optimal control problems are studied through a bibliometric viewpoint.

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