In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinea...In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).展开更多
The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets ...The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented.展开更多
The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For thi...The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.展开更多
In this topic, a new. approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solution of time-var...In this topic, a new. approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solution of time-variation dynamics is therefore established, which is beneficial to further research of time-variation science.展开更多
The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This ...The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.展开更多
文摘In this paper, we present a practical matrix method for solving nonlinear Volterra-Fredholm integro-differential equations under initial conditions in terms of Bernstein polynomials on the interval [0,1]. The nonlinear part is approximated in the form of matrices’ equations by operational matrices of Bernstein polynomials, and the differential part is approximated in the form of matrices’ equations by derivative operational matrix of Bernstein polynomials. Finally, the main equation is transformed into a nonlinear equations system, and the unknown of the main equation is then approximated. We also give some numerical examples to show the applicability of the operational matrices for solving nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs).
文摘The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented.
文摘The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.
文摘In this topic, a new. approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solution of time-variation dynamics is therefore established, which is beneficial to further research of time-variation science.
文摘The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.
