scholarly journals Analysis of Linear Piecewise Constant Delay Systems Using a Hybrid Numerical Scheme

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
H. R. Marzban ◽  
S. M. Hoseini
Keyword(s):  
Numerical Scheme ◽  
Delay Systems ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Constant Delay ◽  
Smooth Functions ◽  
Piecewise Constant ◽  
Delay Function ◽  
Linear Delay

An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
H. R. Marzban ◽  
S. Hajiabdolrahmani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Operational Matrix ◽  
Delay Systems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Constant Delay ◽  
Piecewise Constant ◽  
Delay Function ◽  
Delay Differential

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.

Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Chandrali Baishya ◽  
P. Veeresha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Laguerre Polynomial ◽  
Fractional Integration ◽  
Laguerre Polynomials ◽  
Operational Matrix ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differential

The Atangana–Baleanu derivative and the Laguerre polynomial are used in this analysis to define a new computational technique for solving fractional differential equations. To serve this purpose, we have derived the operational matrices of fractional integration and fractional integro-differentiation via Laguerre polynomials. Using the derived operational matrices and collocation points, we reduce the fractional differential equations to a system of linear or nonlinear algebraic equations. For the error of the operational matrix of the fractional integration, an error bound is derived. To illustrate the accuracy and the reliability of the projected algorithm, numerical simulation is presented, and the nature of attained results is captured in diverse order. Finally, the achieved consequences enlighten that the solutions obtained by the proposed scheme give better convergence to the actual solution than the results available in the literature.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zain ul Abdeen ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Haar Wavelets ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Fractional Differential

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

Analysis of Time-delay Systems via Hybrid of Block-pulse Functions and Taylor Series

2005 ◽  
Vol 11 (12) ◽  
pp. 1455-1468 ◽  
Author(s):  
H. R. Marzban ◽  
M. Razzaghi
Keyword(s):  
Time Delay ◽  
Taylor Series ◽  
Delay Systems ◽  
Operational Matrices ◽  
Time Delay Systems ◽  
Algebraic Equations ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Hybrid Function

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.

scholarly journals Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Guodong Shi ◽  
Yanlei Gong ◽  
Mingxu Yi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Tabular Form ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Singular Kernels

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

A spectral framework for the solution of fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel

2020 ◽  
pp. 107754632097481
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Caputo Fractional Derivative ◽  
Variational Problems ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control

A new fractional-order Dickson functions are introduced for solving numerically fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel. The type of fractional derivative in the proposed problems is the Atangana–Baleanu–Caputo fractional derivative. In the process of the method, we use fractional-order Dickson functions and their properties to provide an accurate computational technique for calculating operational matrices, at first. Then, with the help of operational matrices and the Lagrange multiplier method, these problems are reduced to a system of algebraic equations. At last, to demonstrate the effectiveness of the new method, we enforce the proposed algorithm for several examples.

scholarly journals Discontinuous Legendre Wavelet Galerkin Method for Optimal Control of Time Delayed Systems

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Optimal Control ◽  
Practical Importance ◽  
Delay Systems ◽  
Operational Matrices ◽  
Time Delay Systems ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Delayed Systems ◽  
Problem Of Time ◽  
Delay Differential

Time-delay systems arise in many important applications in science and engineering and optimal control of delay differential equations are of theoretical and practical importance. This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving optimal control problem of time-delayed systems. This new method demonstrates that operational matrices of derivative, delay and product are lower dimensions and sparse because of calculation only on each subinterval. The advantages are implemented to solve algebraic equations transformed from the time-delayed systems with less storage space and execution time. Finally, an experiment is included to illustrate the effectiveness and applicability of the proposed method

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