Operational Matrices Of Integration Research Articles

Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

A new numerical method for delay and advanced integro-differential equations

A general formulation is constructed for Jacobi operational matrices of integration, product, and delay on an arbitrary interval. The main purpose of this study is to improve Jacobi operational matrices for solving delay or advanced integro–differential equations. Some theorems are established and utilized to reduce the computational costs. All algorithms can be used for both linear and nonlinear Fredholm and Volterra integro-differential equations with delay. An error estimator is introduced to approximate the absolute error when the exact solution of a given problem is not available. The error of the proposed method is less compared to other common methods such as the Taylor collocation, Chebyshev collocation, hybrid Euler–Taylor matrix, and Boubaker collocation methods. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.

Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed Volterra–Fredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as N increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the Block–Pulse functions, differential transform, and degenerate kernel methods.

Computational method for one-dimensional heat equation subject to non-local conditions

In this paper, a method based on Bernstein polynomial is proposed for numerical solution of an one-dimensional heat equation subject to non-local boundary conditions. Properties of Bernstein polynomial and operational matrices of integration, differentiation and the product are stated and utilised to reduce the solution of the given elliptic partial differential equation to the solution of the system of algebraic equations. Illustrative numerical tests are given to compare the efficiency and accuracy of the new method with other methods.

Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices

In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

Numerical solution of nonlinear integral equations using alternative Legendre polynomials

In this paper, the operational matrices of integration and the product for the alternative Legendre polynomials (ALPs) are first derived. Then, using these operational matrices and the collocation method, the nonlinear Volterra–Fredholm–Hammerstein integral equations are reduced to a set of nonlinear algebraic equations with unknown ALP coefficients. Some error estimations are provided and the efficiency and accuracy is verified by applying the method to some examples chosen from other literature.

A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems

In this paper‎, two inverse problems of determining an unknown source term in a parabolic‎ equation are considered‎. ‎First‎, ‎the unknown source term is ‎estimated in the form of a combination of Chebyshev functions‎. ‎Then‎, ‎a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem‎. ‎For solving the problem‎, ‎the operational matrices of integration and derivation are introduced and utilized to reduce the mentioned problem into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients‎. Due‎ to ill-posedness of these inverse problems‎, ‎the Tikhonov regularization method with generalized cross validation (GCV) criterion is applied to find stable‎ solutions. ‎Finally‎, some examples are presented to illustrate the efficiency of this numerical method‎. The numerical results show that the proposed method is a reliable method and can give high accuracy approximate solutions.

Parameter identification of linear multi-delay systems via a hybrid of block-pulse functions and Taylor’s polynomials

ABSTRACTIn this paper, an efficient and effective procedure is successfully developed for parameter identification of linear time-invariant multi-delay systems. The proposed framework is based on a hybrid of block-pulse functions and Taylor’s polynomials. Two upper error bounds corresponding to hybrid functions are established. The excellent properties of these functions together with the associated operational matrices of integration and delay are utilised to transform the original problem into a system of linear algebraic equations. The least squares method is then implemented for estimation of the unknown parameters. Several numerical experiments are investigated to demonstrate the usefulness and effectiveness of the proposed procedure. Easy implementation, simple operations and accurate solutions are the main features of the suggested approximation scheme.

Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet

In the current study, new functions called generalized fractional-order Bernoulli wavelet functions (GFBWFs) based on the Bernoulli wavelets are defined to obtain the numerical solution of fractional-order pantograph differential equations in a large interval. For the concept of fractional derivative we will use Caputo sense by using Riemann–Liouville fractional integral operator. First, the generalized fractional-order Bernoulli wavelets are constructed. Then, these functions and their properties are employed to derive the GFBWFs operational matrices of fractional integration and pantograph. The operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations. Finally, some examples are included for demonstrating the validity and applicability of our method.

Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) together with the Galerkin method is proposed for solving a class of nonlinear stochastic Ito–Volterra integral equations. For this purpose, a new stochastic operational matrix (SOM) for LWs is derived. A collocation method based on hat functions (HFs) is employed to derive a general procedure for forming this matrix. The LWs and their operational matrices of integration and stochastic Ito-integration and also some useful properties of these basis functions are used to transform such problems into corresponding nonlinear systems of algebraic equations, which can be simply solved to achieve the solution of such problems. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient. Furthermore as some useful applications, the proposed method is applied to obtain approximate solutions for some stochastic problems in the mathematics finance, biology, physics and chemistry.

Application of operational matrices to numerical solution of stochastic SIR model

The goal of this paper is to give useful method for solving a problem in biologic system that is formulated by stochastic Volterra integral equations. Here, we consider triangular functions, block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the validity and applicability of the operational matrices.

Numerical solutions of stochastic Lotka-Volterra equations via operational matrices

In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.

On the Analysis and Numerical Formulation of Miscible Fluid Flow in Porous Media Using Chebyshev Wavelets Collocation Method

In this paper, the Chebyshev wavelet method, constructed from the Chebyshev polynomial of the first kind is proposed to numerically simulate the single-phase flow of fluid in a reservoir. The method was used together with the operational matrices of integration which resulted in an algebraic system of equations. The system of equation was solved for the wavelet coefficient and used to construct the solutions. The efficiency and accuracy of the method were demonstrated through error measurements. Both the root mean square and the maximum absolute error analysis used in the study were within significantly close range. The Chebyshev wavelet collocation method subsequently was observed to closely approximate the analytic solution to the single phase flow model quite well.

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.

An Efficient Method for Dynamic Analysis of Spatial Trusses Using Legendre Wavelets

This paper introduces a procedure where the free-scaled Legendre wavelets are improved for dynamic analysis of spatial trusses. The operational matrices of integration for free scales of Legendre wavelets are presented and applied for time history analysis. For this aim, a clear-cut formulation is derived from decomposition of the only transitional responses of structures. The free-scaled operational matrices of Legendre wavelets accurately transfer the second-ordered differential equations of motion into the corresponding algebraic equations. Accordingly, a simple procedure is developed to compute dynamic responses of large-scaled spatial trusses. The applicability and effectiveness of the proposed procedure using Legendre wavelets are demonstrated with two numerical applications. Subsequently, results were compared with those from common time integration schemes such as Wilson-θ, Newmark-β, central difference and Duhamel integration methods. It is deduced that because of the inherent shape functions of Legendre wavelets the dynamic analysis is optimally achieved with the highest accuracy, best computational competency and with the lowest storage capacity.

Wavelets Galerkin method for solving stochastic heat equation

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Optimal Control of Time Delay Systems via Hybrid of Block-Pulse Functions and Orthonormal Taylor Series

A new method to find the optimal control of time delay systems with quadratic performance index is discussed. The method is based on hybrid functions. The properties of the hybrid functions which consists of block-pulse functions and orthonormal Taylor series are presented. The operational matrices of integration, delay, dual and product are used to reduce the solution of optimal control time delay system to the solution of algebraic equations. Numerical examples are included to illustrate the effectiveness and validity of the technique.

Optimal control of linear multi‐delay systems based on a multi‐interval decomposition scheme

SummaryThis paper presents a novel approximation scheme to the numerical treatment of linear time‐varying multi‐delay systems with a quadratic performance index. A direct approach based on a hybrid of block‐pulse functions and Chebyshev polynomials is successfully developed. The operational matrix of delay associated to multi‐delay systems is constructed by an efficient manner. The excellent properties of hybrid functions together with the operational matrices of integration, delay, and product are then used to transform the optimal control problem into a mathematical optimization problem whose solution is much more easier than the original one. The procedure described in the current paper can be regarded as a multi‐interval decomposition scheme. The convergence of the proposed method is verified numerically. A wide variety of multi‐delay systems are investigated to demonstrate the effectiveness and computational efficiency of the proposed numerical scheme. The method has a simple structure, is easy to implement, and provides very accurate solutions. Copyright © 2015 John Wiley & Sons, Ltd.

Spectral method for solving high order nonlinear boundary value problems via operational matrices

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.