Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

Identification of nonlinear systems using hybrid functions

Most real systems have nonlinear behavior and thus model linearization may not produce an accurate representation of them. This paper presents a method based on hybrid functions to identify the parameters of nonlinear real systems. A hybrid function is a combination of two groups of orthogonal functions: piecewise orthogonal functions (e.g. Block-Pulse) and continuous orthogonal functions (e.g. Legendre polynomials). These functions are completed with an operational matrix of integration and a product matrix. Therefore, it is possible to convert nonlinear differential and integration equations into algebraic equations. After mathematical manipulation, the unknown linear and nonlinear parameters are identified. As an example, a mechanical system with single degree of freedom is simulated using the proposed method and the results are compared against those of an existing approach.<br /

An efficient hybrid pseudo-spectral method for solving optimal control of Volterra integral systems

In this paper, a new pseudo-spectral (PS) method is developed for solving optimal controproblems governed by the non-linear Volterra integral equation(VIE). The novel method is based upon approximating the state and control variables by the hybrid of block pulse functions and Legendre polynomials. The properties of hybrid functions are presented. The numerical integration and collocation method is utilized to discretize the continuous optimal control problem and then the resulting large-scale finite-dimensional non-linear programming (NLP) is solved by the existing well-developed algorithm in Mathematica software. A set of sufficient conditions is presented under which optimal solutions of discrete optimal control problems converge to the optimal solution of the continuous problem. The error bound of approximation is also given. Numerical experiments confirm efficiency of the proposed method especially for problems with non-sufficiently smooth solutions belonging to class $C^1$ or $C^2$

A hybrid functions method for solving linear and non-linear systems of ordinary differential equations

In the present paper, we use a hybrid method to solve linear or non-linear systems of ordinary differential equations (ODEs). By using this method, these systems are reduced to a linear or non-linear system of algebraic equations. In error discussion of the suggested method, an upper bound of the error is obtained. Also, to survey the accuracy and the efficiency of the present method, some examples are solved and comparisons between the obtained results with those of several other methods are carried out

Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework

An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique

A novel Chebyshev wavelet method for solving fractional-order optimal control problems

This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature