Operational Matrices Of Derivative Research Articles

An efficient wavelet‐based approximation method for solving nonlinear fractional‐time long wave equations: An operational matrix approach

This research paper develops an efficient Gegenbauer wavelet‐based approximation method to solve nonlinear fractional‐time regularized long wave (RLW) equations arising in ocean engineering. The operational matrices of derivatives, as well as the fractional order integration, are engaged in the proposed method. Using the operational matrix method and collocation point, the nonlinear RLW equations are converted into a system of algebraic equations, and these equations are further solved. Some results regarding the error‐bound estimation of this method have been developed. The method's accuracy and efficiency are confirmed by error estimation. The obtained results are compared with other available results and exact solutions. Moreover, the use of Gegenbauer wavelets is found to be a simple, straightforward forward and efficient tool for solving nonlinear PDEs arising in ocean engineering.

A collocation method using generalized Laguerre polynomials for solving nonlinear optimal control problems governed by integro-differential equations

The Laguerre polynomials represent a new set of basic functions adopted to solve a class of optimal control problems (OCP) specified by integro-differential equation namely IOCP. The corresponding operational matrices of derivatives are computed to extend the solution of the problem in terms of Laguerre polynomials. Using the results of the considered basis functions with a joint application of collocation method, the IOCP is reduced to solve a system of nonlinear algebraic system. Come acrossing an error bound and convergence analysis for the approximate optimal value of the performance index have been proved by the proposed method. Finally, some examples are given to bring about the validity and applicability of this technique.

Parametric Identification for the Biased Ship Roll Motion Model Using Genocchi Polynomials

Roll motion is one of the key motions related to a vessel’s dynamic stability. It is essential for the dynamic stability of ships in the realistic sea. For this research study, we have investigated the parameters involved in damping of the ship. In general, mathematical modelling of the rolling response of a ship can be formulated by the linear, nonlinear, and fractional differential equations because the amplitude of oscillation is increased. An efficient Genocchi polynomial approximation method (GPAM) is successfully applied for the biased ship roll motion model. The basic idea of the collocation method together with the operational matrices of derivatives used for nonlinear differential equation and convert it into a system of algebraic equations. The convergence and error analysis of the proposed method are also discussed. A few numerical experiments are carried out for some specific and important types of problems including the biased roll motion equations. The results are compared to those produced using the Legendre wavelet method (LWM) and the homotopy perturbation method (HPM). It is observed that the proposed spectral algorithm is robust, accurate, and easy to apply.

New operational matrices approach for optimal control based on modified Chebyshev polynomials


 
 
 
 The purpose of this paper is to introduce interesting modified Chebyshev orthogonal polynomial. Then, their new operational matrices of derivative and integration or modified Chebyshev polynomials of the first kind are introduced with explicit formulas. A direct computational method for solving a special class of optimal control problem, named, the quadratic optimal control problem is proposed using the obtained operational matrices. More precisely, this method is based on a state parameterization scheme, which gives an accurate approximation of the exact solution by utilizing a small number of unknown coefficients with the aid of modified Chebyshev polynomials. In addition, the constraint is reduced to some algebraic equations and the original optimal control problem reduces to optimization technique, which can be solved easily, and the approximate value of the performance index is calculated. Moreover, special attention is presented to discuss the convergence analysis and an upper bound of the error for the presented approximate solution is derived. Finally, some important illustrative examples of obtained results are shown and proved that powerful method in a simple way to get an optimal control of the considered.
 
 
 

Fibonacci polynomials for the numerical solution of variable‐order space‐time fractional Burgers‐Huxley equation

In this article, the variable‐order (VO) space‐time fractional version of the Burgers‐Huxley equation is introduced with fractional differential operator of the Caputo type. The collocation technique based on the Fibonacci polynomials (FPs) is developed for finding the approximate solution of this equation. In order to implement the presented method, some novel operational matrices of derivative (including ordinary and fractional derivatives) are extracted for the FPs. Moreover, the roots of the Chebyshev polynomials of the first kind are chosen as the collocation points which reduce the equation to a system of algebraic equations more efficiency. Ultimately, we obtain the solution of the VO space‐time fractional Burgers‐Huxley equation in terms of the FPs. The devised method is validated by finding an error bound for the truncated series of the Fibonacci expansion in two dimensions. The accuracy of approximation is verified through various illustrative examples.

Direct method for solution variational problems by using Hermite polynomials

‎In this approach‎, ‎one‎ computational method is presented for numerical approximation of variational problems‎. ‎This method with variable ‎coeffici‎ents is based on Hermite polynomials‎. ‎The properties of Hermite polynomials with the operational matrices of derivative and integration are used to reduce optimal control problems to the solution of linear algebraic equations‎. ‎Illustrative examples are included to demonstrate the validity and applicability of the technique‎.

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

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

Linearized stable spectral method to analyze two‐dimensional nonlinear evolutionary and reaction‐diffusion models

AbstractThe work is devoted to the development of a new spectral method based on higher dimensional orthogonal polynomials. Firstly, the concept of traditional Chelyshkov polynomials is extended for the function of more than one variable while definitions and theorems are presented with proofs. The operational matrices of derivative have been constructed assisted by defined higher‐order polynomials and used to the development of a spectral method. The method is further coupled with a Picard‐iterative scheme to tackle the high nonlinearities and termed as the Picard–Chelyshkov polynomial method (PCPM). The convergence and error‐bound have been analyzed through theorems and their proofs in order to prove the mathematical authenticity of the method. The PCPM is applied for some two‐dimensional unsteady nonlinear fractional partial differential equations and efficient results have been attained. In addition, it is evident from the comparative analysis with existing literature that the proposed method is fair enough in terms of accuracy, efficiency, and cost to deal with the problem in higher dimensions whilst could be further modified for other classes of dynamical problems.

Solving a Quadratic Riccati Differential Equation, Multi-Pantograph Delay Differential Equations, and Optimal Control Systems with Pantograph Delays

An effective algorithm for solving quadratic Riccati differential equation (QRDE), multipantograph delay differential equations (MPDDEs), and optimal control systems (OCSs) with pantograph delays is presented in this paper. This technique is based on Genocchi polynomials (GPs). The properties of Genocchi polynomials are stated, and operational matrices of derivative are constructed. A collocation method based on this operational matrix is used. The findings show that the technique is accurate and simple to use.

A new hybrid method for two dimensional nonlinear variable order fractional optimal control problems

AbstractThis paper presents an efficient operational matrix method for two dimensional nonlinear variable order fractional optimal control problems (2D‐NVOFOCP). These problems include the nonlinear variable order fractional dynamical systems (NVOFDS) described by partial differential equations such as the diffusion‐wave, convection‐diffusion‐wave and Klein‐Gordon equations. The variable order fractional derivative is defined in the Caputo type. The proposed hybrid method is based on the transcendental Bernstein series (TBS) and the generalized shifted Chebyshev polynomials (GSCP). The new operational matrices of derivatives are generated for the mentioned polynomials. The state and control functions are expressed by the TBS and GSCP with free coefficients and control parameters. These expansions are substituted in the performance index and the resulting operational matrices are employed to extract algebraic equations from the approximated NVOFDS. The constrained extremum is obtained by coupling the algebraic constraints from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the performance index by means of a set of unknown Lagrange multipliers. The convergence analysis is discussed and several numerical experiments illustrate the efficiency and accuracy of the proposed method.

Solving Optimal Control Problems by using Hermite polynomials

‎In this paper‎, ‎one numerical method is presented for numerical approximation of linear constrained optimal control problems with quadratic performance index‎. ‎The method with variable coefficients is based on Hermite polynomials‎. ‎The properties of Hermite polynomials with the operational matrices of derivative are used to reduce optimal control problems to the solution of linear algebraic equations‎. ‎Illustrative examples are included to demonstrate the validity and applicability of the technique‎.

A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model

The evolution equations with fractional or variable order derivatives can deliver a proper mathematical modeling to define the transport dynamics and anomalous diffusion in complex dynamical structures. Herein, a hybrid method based on operational matrices of derivative is proposed and successfully applied to explore the solution of mobile–immobile advection–dispersion problem of variable order. The variable order of the model is considered as function of space and time. The operational matrices of derivative named exact and approximate are constructed with the aid of two different approaches and related theorems are available to support the mathematical justification. The error bound and convergence analysis is presented to validate the mathematical formulation of the computational algorithm. A comparative study is enclosed in our investigation which endorses the credibility of the exact operational matrix of derivative. The numerical simulations for various problems are encountered and set of graphs are presented. The numerical examples are endorsing that the proposed mathematical algorithm is computationally effective and efficient tool and one can extend it to other physical problems of fractional or variable order.

On the Properties of Two-Dimensional Normalized Boubaker Polynomials

The aim of the present work deals with newly defined two-variable polynomials for normalized Boubaker . The operational matrices of derivatives with respect to the two variables are presented at first with explicit expression. Then, a normalized Boubaker polynomial approximation for the numerical solution of a class of partial differential equations is proposed, depending on a truncated, normalized Boubaker function series in the equation together with the operational matrices in the proposed partial differential equation. The original partial differential equation is reduced under consideration of a system of simply solvable algebraic equations. Due to the interesting derived properties of normalized Boubaker polynomials in two variables, the suggested method can achieve good results with few complexities. Using operational matrices of derivatives, one can save computation and more memory. Two-dimensional examples are listed to show the satisfactory level of the suggested method.

Exponential Solution for the Natural Convection of a Darcian Fluid About a Full Cone in a Porous Medium

In the current work, a new approach based on exponential functions is presented to reach the exact solutions to the problem of natural convection about a full cone which is set in a porous medium for the both states of prescribed surface heat flux and wall temperature. The solutions are described in the form of the exponential functions series with unknown coefficients while the operational matrices of derivative and product of these functions are utilized to find these coefficients iteratively without solving the system of nonlinear equations. The solutions demonstrate the validity and applicability of the proposed technique by means of the residual function error norm.

Wavelet-Based Approximation Algorithms for Some Nonlinear Oscillator Equations Arising in Engineering

In this paper, we have applied the wavelet-based computational algorithms to solve some nonlinear oscillator equations arising in engineering. Operational matrices of derivatives are used to convert nonlinear differential equation into a system of linear/nonlinear algebraic equations. Convergence analysis of the wavelet algorithms is discussed. The obtained results are validated with the results obtained by homotopy perturbation method (HPM), modified homotopy perturbation method, differential transformation method (DTM), modified differential transform method and MATLAB solutions. Satisfactory agreement with the HPM, DTM and numerical solution is observed.

Generalized shifted Chebyshev polynomials for fractional optimal control problems

The generalized shifted Chebyshev polynomials (GSCP) represent a novel class of basis functions that include free coefficients and control parameters. The GSCP are adopted to solve a class of fractional optimal control problems (FOCP). The corresponding operational matrices of derivatives are calculated for expanding the solution of the problem in terms of the GSCP. Numerical examples show the good performance of the algorithm.

Analytical expressions of amperometric enzyme kinetics pertaining to the substrate concentration using wavelets

In this paper, wavelet-based operational matrix methods have been developed to investigate the approximate solutions for amperometric enzyme kinetics problems. The operational matrices of derivatives have been utilized for solving the nonlinear initial value problems. The accuracy of the proposed wavelet-based approximation methods has been confirmed. The main purpose of the proposed method is to get better and more accurate results. Operational matrices of Chebyshev and Legendre wavelets are utilized to obtain a sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of nonlinear initial value problems. Numerical experiments are given to demonstrate the accuracy and efficiency of the proposed method.

A new approach based on the Zernike radial polynomials for numerical solution of the fractional diffusion-wave and fractional Klein–Gordon equations

This study aims to present some new numerical approximation schemes based on operational matrices of the Zernike radial polynomials (ZRPs), which are employed in solving fractional diffusion waves and fractional Klein–Gorden equations in fluid mechanics. First, some definitions of fractional calculus are given along some of the ZRPs and their operational matrices of derivative and fractional Riemann–Liouville integration. The main advantage of this approach is the reduction of the fractional diffusion-wave and fractional Klein–Gordon equation to the algebraic equations systems that can be solved easily, thus simplifying the problem. An estimation of the error is given. Finally, illustrative instances are discussed to shed more light on the validity and applicability of the presented method.

Hermite interpolant multiscaling functions for numerical solution of the convection diffusion equations

A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diusion equation to the solution of algebraic equations. Because of sparsity of these matrices, this method is computationally very attractive and reduces the CPU time and computer memory. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Analytical Expressions Pertaining to the Steady State Concentrations of Glucose, Oxygen and Gluconic Acid in a Composite Membrane Using Genocchi Polynomials

In this paper, a new analytical method is reported for the steady state concentrations of glucose, oxygen and gluconic acid in a sensitive composite membrane that contains nanoparticles of an anionic polymer glucose oxidase and catalyse embedded in a hydrophobic polymer. The theoretical model involves a system of nonlinear coupled reaction–diffusion equations. This polynomial approximation method is based on the Genocchi polynomials together with their operational matrices of derivative. Also the convergence analysis of the proposed method is discussed. The obtained GPM solutions are compared with the Adomian decomposition method (ADM) solutions. Satisfactory agreement with ADM and numerical results is observed. Moreover, the use of GPM is found to be simple, efficient, flexible and less computation costs.