Delay Operational Matrix Research Articles

Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann–Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann–Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

Numerical simulation of variable-order fractal-fractional delay differential equations with nonsingular derivative

This work develops a new Legendre delay operational matrix based on Legendre polynomial features that are integrated with regard to the Legendre fractional derivative operational matrix in order to solve the issues. The motivation behind solving the Atangana-Baleanu The variable-order fractal-fractional delay differential equations rely on the properties of the kernel in the Atangana-Baleanu fractal-fractional derivative operator. Atangana-Baleanu fractal-fractional derivative by the variable-order exponential kernel gives more precise results to the derivative. The Legendre operational matrix of the fractional derivative error bound is also shown here. The variable-order fractal-fractional delay differential equations with Atangana-Baleanu derivatives are reduced to a set of algebraic equations using a collocation strategy based on these operational matrices. The numerical findings show that the proposed approach is a useful mathematical tool for calculating numerical solutions to variable-order fractal-fractional delay differential equations with an Atangana-Baleanu derivative compared to earlier techniques. At last, the numerical examples are employed to show the performance and efficiency of the method.

Parameter identification of fractional-order time delay system based on Legendre wavelet

This paper proposes a parameter identification method of fractional-order time delay system based on Legendre wavelet. An integration operational matrix and delay operational matrix of a Legendre wavelet are constructed. The process is derived to convert a fractional-order time delay system based on a Legendre wavelet operational matrix to algebraic equation. The least squares method is used to simultaneously estimate the order, model parameters and time delay coefficient of a system. In addition, wavelet decomposition and reconstruction algorithm are used to pre-process the output signal, which separate the noise from signal. The proposed method overcomes the initial-state dependence and randomness of the identification result in the algebraic solution of the traditional operational matrix, and reduces the effect of noise on the accuracy of parameter identification. It is especially suitable for engineering application. The effectiveness is verified by numerical simulation and a temperature process control system.

General Lagrange-hybrid functions and numerical solution of differential equations containing piecewise constant delays with bibliometric analysis

Delay differential equations have been the topic of significant interest in scientific and engineering problems. In this manuscript, we deal with the numerical solution of delay differential equations containing piecewise constant delays. The present method is based on general Lagrange-hybrid functions (GLHFs) and collocation method. First, we define GLHFs without considering the nodes of Lagrange polynomials. The integration operational matrix and delay operational matrix of GLHFs are obtained, generally. Using the integration and delay operational matrices and collocation method, the proposed problem transforms into a system of algebraic equations. An estimation of the error is derived in the sense of the Sobolev norm. Several numerical examples are given to demonstrate the accuracy and validity of the proposed computational procedure, as the mathematical model of tumor growth in mice and HIV infection of CD4+ T-cells. Moreover, delay differential equations are studied through a bibliometric viewpoint.

An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.

Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives

Abstract The main idea of this paper is to establish the novel fractional Gegenbauer functions (FGFs) for solving three kinds of fractional differential equations generated by the variable-order fractional derivatives in the Atangana-Baleanu-Caputo (ABC) sense. The numerical scheme is discussed based on the modified operational matrices (MOMs) of Atangana-Baleanu variable-order (AB-VO) fractional integration and the delay operational matrix. The methodology of obtaining the MOMs of integration is calculated with high accuracy. So that the precision of the computation method is influenced directly by the proposed matrix. In addition, we investigate the error analysis of the proposed approach. At last, several numerical experiments are employed to clarify the performance and efficiency of the method.

An approximate approach for the generalized variable-order fractional pantograph equation

This study advances variable-order (VO) fractional delay differential models in the pantograph type introduced in the Caputo sense. A method utilizing the Chebyshev cardinal functions (CCFs) is formulated to find an accurate result. In the proposed scheme, we first expand the solution in terms of the CCFs including unknown coefficients. Then, by inserting this expansion into the problem, employing the VO fractional differentiation operational matrix (OM) and the delay OM, we construct a system with algebraic equations. The generated algebraic systems are mainly sparse due to the cardinality of these functions and it significantly reduces the cost of computation.

Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials

Accurate solutions of nonlinear multi-dimensional delay problems of fractional-order arising in mathematical physics and engineering recently have been found to be a challenging task for the research community. This paper witnesses that an efficient fully spectral operational matrices-based scheme is developed and successfully applied for stable solutions of time-fractional delay differential equations (DDEs). Monomials are introduced in order to proposed the novel operational matrices for fractional-order integration Iν and derivative Dν by means of shifted Gegenbauer polynomials. Some ordinary and partial delay differential equations of fractional-order are considered to show reliability, efficiency and appropriateness of the proposed method. In order to approximate the delay term in DDEs a novel delay operational matrix Θba is introduced with the help of shifted Gegenbauer polynomials. The proposed algorithm transform the problem understudy into a system of algebraic equations which are easier to tackle. Analytical solutions of the mentioned problem are effectively obtained, and an inclusive comparative study is reported which reveals that the proposed computational scheme is effective, accurate and well-matched to investigate the solutions of aforementioned problems. Error bound analysis is enclosed in our investigation to reveal the consistency and support the mathematical formulation of the algorithm. This proposed scheme can be extended to explore the solution of more dervisfy problem of physical nature in complex geometry.

A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions

In the current study, a new technique for solving various kinds of fractional partial differential equations based upon Genocchi hybrid functions (GHFs) is projected. Genocchi hybrid collocation technique by utilizing delay operational matrix, integer and the fractional operational matrix of the derivative is introduced. We calculate the delay operational matrix without error, this matrix is a reason to achieve the approximate solution with high accuracy. These operational matrices are applied to convert the proposed equations to a set of algebraic equations with unknown GHFs coefficients. Also, we investigate the error analysis based on Sobolev space. At last, several examples are examined with graphs and tables to demonstrates the efficiency and accuracy of the numerical plan.

Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations

The aim of the current paper is to propose an efficient method for finding the approximate solution of fractional delay differential equations. This technique is based on hybrid functions of block-pulse and fractional-order Fibonacci polynomials. First, we define fractional-order Fibonacci polynomials. Next, using Fibonacci polynomials of fractional-order, we introduce a new set of basis functions. These new functions are called fractional-order Fibonacci-hybrid functions (FFHFs) which are appropriate for the approximation of smooth and piecewise smooth functions. The Riemann–Liouville integral operational matrix and delay operational matrix of the FFHFs are obtained. Then, using these matrices and collocation method, the problem is reduced to a system of algebraic equations. Using Newton’s iterative method, we solve this system. Some examples are proposed to test the efficiency and effectiveness of the present method. Given the application of these kinds of fractional equations in the modeling of many phenomena, a numerical example of this work includes the Hutchinson model which describes the rate of population growth.

Fractional-order Lagrange polynomials: An application for solving delay fractional optimal control problems

The main purpose of this work is to provide an efficient method for solving delay fractional optimal control problems (DFOCPs). Our method is based on fractional-order Lagrange polynomials (FLPs) and the collocation method. The FLPs are used to achieve a new operational matrix of fractional derivative. Also, we present a delay operational matrix of FLPs. These operational matrices are driven without considering the nodes of Lagrange polynomials. The operational matrices and collocation method are applied to a constrained extremum in order to minimize the performance index. Then, the problem reduces to the solution of a system of algebraic equations. Convergence of the algorithm and approximation of FLPs are proposed. Furthermore, the upper bound of the error for the operational matrix of fractional derivatives is obtained. Numerical tests for demonstrating the efficiency and effectiveness of the method are included. Moreover, the method is used for numerical solution of a mathematical model of chemotherapy in breast cancer.

Legendre Wavelets with Scaling in Time-delay Systems

This research presents the integration, product, delay and inverse time operational matrices of Legendre wavelets with an arbitrary scaling parameter and illustrates how to design this parameter in order to improve their accuracy and capability in handling optimal control and analysis of time-delay systems. Using the presented Legendre wavelets, the piecewise delay operational matrix is derived to develop the applicability of Legendre wavelets in systems with piecewise constant time-delays or time-varying delays. With the aid of these matrices, the new Legendre wavelets method is applied on linear time-delay systems. The reliability and efficiency of the method are demonstrated by some numerical experiments.

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.

A novel approach of fractional-order time delay system modeling based on Haar wavelet

In this paper, fractional-order time delay system modeling is presented using Haar wavelet operational matrix of integration. Therefore, it does not require any prior knowledge of transfer function structure or partial information about fractional differentiation order. It allows the estimation of the implicit time delay parameter together with other model parameters by utilizing new delay operational matrix of Haar wavelet based on Riemann-Liouville definition. The proposed technique reduces the complexity of identification by converting the complex fractional calculus equations into simple algebra. The efficacy of the approach is verified on various integer and non-integer (fractional) order systems in simulation. For realistic condition, proposed method is verified in the presence of noise in simulation and also demonstrated on the real-time process control temperature system.

Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis

In the present paper, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet basis and their properties are utilized to obtain their operational matrix of fractional integration in the Riemann–Liouville sense and delay operational matrix. Convergence and error bound of the expansion by this kind of wavelet functions are investigated. Then, these operational matrices along with the Galerkin approach have been implemented to solve systems of fractional delay differential equations (SFDDEs). The main superiority of the proposed technique is that it reduces SFDDEs to a system of algebraic equations. Moreover, accuracy and efficiency of the suggested Chelyshkov wavelet approach are verified through some linear and nonlinear SFDDEs. Finally, the obtained numerical results are compared with those previously reported in the literature.

Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems

In this paper, a new set of functions called fractional-order Boubaker functions is defined for solving the delay fractional optimal control problems with a quadratic performance index. To solve the problem, first we obtain the operational matrix of the Caputo fractional derivative of these functions and the operational matrix of multiplication to solve the nonlinear problems for the first time. Also, a general formulation for the delay operational matrix of these functions has been achieved. Then we utilized these matrices to solve delay fractional optimal control problems directly. In fact, the delay fractional optimal control problem converts to an optimization problem, which can then be easily solved with the aid of the Gauss–Legendre integration formula and Newton’s iterative method. Convergence of the algorithm is proved. The applicability of the method is shown by some examples; moreover, a comparison with the existing results shows the preference of this method.

Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

Analysis of linear time-invariant time-delay systems via orthogonal functions

This paper first critically reviews the up-to-date literature available on the problem of analysis of linear time-invariant dynamical systems containing time delay(s) and discusses the practical limitations and shortcomings of existing orthogonal functions techniques. Then it presents some mathematical preliminaries very important for the analysis of delay systems. As a part of this, it introduces: the derivative operational matrix of Legendre, Laguerre, Hermite and Fourier orthogonal systems; the delay-integration operational matrix of orthogonal polynomial systems; a new integration operational matrix of Fourier orthogonal system; and the error analysis of arbitrary function in terms of these orthogonal functions and some approximate tools. Next, it proposes delay operational matrix approach and time-partition technique for the analysis of delay systems. It shows that recursive algorithms are possible if block pulse functions are used in the proposed methods whose practical limitations are also pointed out. At the end of the paper the analogy between the well known single-term PCBF approach and the proposed time-partition technique is given.

Analysis of time-delay systems via shifted Chebyshev polynomials of the first and second kinds

A novel and general approach for obtaining the delay operational matrix of shifted Chebyshev polynomials of first or second kind is presented. This operational matrix is exact in the sense that it does not involve any approximation. Next, this paper shows the application of the delay operational matrix in the analysis of time-delay systems. Two illustrative examples are included and the results are compared with those obtained by the exact method.

Analysis and parameter identification of time-delay systems via shifted Jacobi polynomials

The shifted Jacobi delay operational matrix is introduced, and applied to the approximation of the solutions of linear systems with arbitrary time delay. The parameter-identification problem for time-delay systems is also dealt with. The new method is algebraic and computer-oriented. Two examples are given, and the results are very accurate and satisfactory.