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In this paper, a new numerical method for solving fractional optimal control problems by using hybrid functions is presented. The Riemann–Liouville fractional integral operator for hybrid functions is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well-developed algorithms may be applied. The method is computationally very attractive and gives very accurate results. [ABSTRACT FROM AUTHOR]

This paper presents a computational technique for the solution of a class of nonlinear singular boundary value problems arising in physiology. The approach is based on a pseudospectral Legendre method in which Legendre-Gauss-Lobatto nodes are used to construct the Nth degree polynomial interpolation to approximate the solution. The method is easy to implement and yields accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature. [ABSTRACT FROM AUTHOR]

This article presents a numerical method for solving nonlinear mixed Volterra-Fredholm integral equations. The method is based on two-dimensional Legendre wavelet approximations. The properties of Legendre wavelets are utilized together with the Gaussian integration method to reduce the mixed integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

A numerical method for finding the solution of linear time-varying differential equations is proposed. The properties of the hybrid functions, which consist of block-pulse functions plus rationalized Haar functions, are presented. The properties of the hybrid functions together with the operational matrices of integration and product are then utilized to reduce the solution of differential equations to the solution of algebraic equations. Examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

In this paper we present a method for finding the solution of time-delay systems using a hybrid function. We present the properties of the hybrid functions, which consist of block-pulse functions plus Taylor series. The method is based upon expanding various time functions in the system as their truncated hybrid functions. Operational matrices of integration and delay are presented and are utilized to reduce the solution of time-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined, and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical method for solving fractional optimal control problems (FOCPs). The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon Bernoulli polynomials. The operational matrices of fractional Riemann–Liouville integration and multiplication for Bernoulli polynomials are derived. The error upper bound for the operational matrix of the fractional integration is also given. The properties of Bernoulli polynomials are utilized to reduce the given optimization problems to the system of algebraic equations. By using Newton’s iterative method, this system is solved and the solution of FOCPs are achieved. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]