State-control parameterization method based on using hybrid functions of block-pulse and Legendre polynomials for optimal control of linear time delay systems

In this paper, a method based on using hybrid functions of block-pulse and Legendre polynomials for finding the optimal solution of systems with delay in state and control variables is presented. The state-control parameterization method is used to convert the original optimal control problem with time delays into an optimization problem. This method does not require operational matrices of delay, product and integration of hybrid functions for obtaining this goal. The validity of this method is examined by illustrative examples.     

Hybrid functions approach for optimal control of systems described by integro-differential equations

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Hybrid functions approach for nonlinear constrained optimal control problems

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Numerical solution of the controlled Duffing oscillator by hybrid functions

A numerical method for solving the controlled Duffing oscillator is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are given. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrix of product is introduced. This matrix together with the operational matrix of integration are utilized to reduce the solution of controlled Duffing oscillator to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Numerical solution of time-varying delay systems by Chebyshev wavelets

The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated Chebyshev wavelets. The operational matrix of delay is introduced. The operational matrices of integration and delay are utilized to reduce the solution of time-varying delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A hybrid approximation method for solving Hutchinson’s equation

The hybrid function approximation method for solving Hutchinson’s equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre-Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson’s equation.     

Analysis of time-varying delay systems via triangular functions

This paper presents a method for finding the analysis of time-varying delay systems using triangular functions. We present the properties of the triangular functions. The operational matrices of integration, delay and product are utilized to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples is included to demonstrate the validity and applicability of the technique.     

A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.     

Solution of Volterra's population model via block‐pulse functions and Lagrange‐interpolating polynomials

A 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. Copyright © 2008 John Wiley & Sons, Ltd.     

A composite Chebyshev finite difference method for nonlinear optimal control problems

In this paper, a composite Chebyshev finite difference method is introduced and is successfully employed for solving nonlinear optimal control problems. The proposed method is an extension of the Chebyshev finite difference scheme. This method can be regarded as a non-uniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev–Gauss–Lobatto points. The convergence of the method is established. The nice properties of hybrid functions are then used to convert the nonlinear optimal control problem into a nonlinear mathematical programming one that can be solved efficiently by a globally convergent algorithm. The validity and applicability of the proposed method are demonstrated through some numerical examples. The method is simple, easy to implement and yields very accurate results.     

Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets

This paper introduces the application of linear Legendre multiwavelets to the optimal control synthesis for linear time-delayed systems. Based on some useful properties of linear Legendre multiwavelets, integration, product and delay operational matrices are proposed to solve the linear time-delayed systems first. Then, a quadratic cost functional is approximated by those properties. By using Lagrange multipliers, the quadratic cost functional is minimized subject to the solution of the linear time-delayed system and an explicit formula for the optimal control is obtained. The effectiveness of the method and accuracy of the solution are shown in comparison with some other methods by illustrative examples.     

Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Numerical solution of the fractional Bagley‐Torvik equation by using hybrid functions approximation

In this paper, a new numerical method for solving the fractional Bagley‐Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block‐pulse functions and Bernoulli polynomials are presented. The Riemann‐Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley‐Torvik differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

Study on convergence of hybrid functions method for solution of nonlinear integral equations

In this article, we present the uniform convergence analysis and accuracy estimation of hybrid functions (HFs) method for finding the solution of nonlinear Volterra and Fredholm integral equations. The properties of HFs which consist of block-pulse functions (BPFs) and Legendre polynomials are used to reduce the solution of nonlinear integral equations to the solution of algebraic equations. The superiority and accuracy of the HFs method to BPF and Legendre polynomial methods are illustrated through some numerical examples.     

Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models

A new efficient type of Chebyshev wavelet is used to find the optimal solutions of general linear, continuous-time, multi-delay systems with quadratic performance indices and also to obtain the responses of linear time-delay systems. According to the new definition of Chebyshev wavelets, the operational matrices of integration, product, delay and inverse time and the integration matrix are derived. Furthermore, new operational matrices as the piecewise delay operational matrix and the stretch operational matrix of the desired Chebyshev wavelets are introduced to analyze systems with, in turn, piecewise constant delays and stretched arguments or proportional delays. Two novel algorithms based on newly Chebyshev wavelet method are proposed for the optimal control and the analysis of delay models. Some examples are solved to establish that the accuracy and applicability of Chebyshev wavelet method in delay systems are increased.     

Exact and approximation product solutions form of heat equation with nonlocal boundary conditions using Ritz–Galerkin method with Bernoulli polynomials basis

In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz–Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz–Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1143–1158, 2017     

Solution of Lane–Emden type equations using rational Bernoulli functions

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane‐Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions

Numerical Algorithms - In this paper, we propose an efficient numerical method based on two-dimensional hybrid of block-pulse functions and Legendre polynomials for numerically solving an inverse...     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.