An initial value problem with piecewise constant coe cients is considered. The accuracies for both nite di erence methods and the pseudospectral method are analyzed, and a modi cation of the initial value problem is suggested. The modi ed problem is shown to have the same temporal period as the original problem does, and a second order accuracy is obtained for the pseudospectral method at integ...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

Abstract— A new polynomial method to solve Volterra–Fredholm Integral equations is presented in this work. The method is based upon Shifted Legendre Polynomials. The properties of Shifted Legendre Polynomials and together with Gaussian integration formula are presented and are utilized to reduce the computation of Volterra–Fredholm Integral equations to a system of algebraic equations. Some num...

This paper considers the numerical simulation of incompressible viscous fluid flow in an infinite strip. A mixed spectral method is proposed using the Legendre approximation in one direction and the Legendre rational approximation in another direction. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Legendre-Legendre rational approximation are establishe...

properties of the hybrid of block-pulse functions and lagrange polynomials based on the legendre-gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known legendre interpolation operator. the uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. the applicabili...

A numerical method based on Legendre multi-wavelets is applied for solving Lane-Emden equations which form Volterra integro-differential equations. The Lane-Emden equations are converted to Volterra integro-differential equations and then are solved by the Legendre multi-wavelet method. The properties of Legendre multi-wavelets are first presented. The properties of Legendre multi-wavelets are ...

In [E. Babolian, M.M. Hosseini, Reducing index, and pseudospectral methods for differential–algebraic equations, Appl. Math. Comput. 140 (2003) 77–90] a reducing index method has been proposed for some cases of semi-explicit DAEs (differential algebraic equations). In this paper, this method is generalized to more cases. Also, it is focused on Hessenberg index 2 systems and proposed reduction i...

In this paper Legendre integral method is proposed to solve optimal control problems governed by higher order ordinary differential equations. Legendre approximat ion method reduced the problem to a constrained optimizat ion problem. Penalty partial quadratic interpolation method is presented to solve the resulting constrained optimization problem. Error estimates for the Legendre approximation...