SUMMARYA numerical method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro-differential equation where the integral term represents the effects of toxin. The approach is based on hybrid function approximations. The properties of hybrid functions that consist of block-pulse and Lagrange-interpolating polynomials are presented. The associated operational matrices of integration and product are then utilized to reduce the solution of Volterra's model to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. Applications are demonstrated through an illustrative example.

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SUMMARYIn this paper, a composite Chebyshev finite difference method is introduced and applied for finding the solution of optimal control of time-delay systems with a quadratic performance index. This method is an extension of the Chebyshev finite difference scheme. The proposed method can be regarded as a nonuniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev-Gauss-Lobatto points. Various types of time-delay systems are included to demonstrate the validity and the applicability of the technique. The method is easy to implement and provides very accurate results.

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Summary This paper presents a composite Chebyshev finite difference method to numerically solve nonlinear optimal control problems with multiple time delays. The proposed discretization scheme is based on a hybrid of block‐pulse functions and Chebyshev polynomials using the well‐known Chebyshev Gauss–Lobatto points. Our approach is an extension and also a modification of the Chebyshev finite difference scheme. A direct approach is used to transform the delayed optimal control problem into a nonlinear programming problem whose solution is much more easier than the original one. Some useful error bounds are established. In addition, the convergence of the method is discussed. A wide variety of numerical experiments are investigated to show the usefulness and effectiveness of the proposed discretization procedure. The method has a simple structure and can be implemented without too much effort. Copyright © 2015 John Wiley & Sons, Ltd.