A Tau method with perturbed boundary conditions for certain ordinary differential equations

Ortiz' recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) aperturbed version of the given differential equation, and (ii) the imposed supplementary conditionsexactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions areperturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proprtional to a Chebyshev polynomial. Dedicated to Eduardo L. Ortiz on the occasion of his 70th birthday     

The weighting subspaces of the Tau Method and orthogonal collocation

In this paper we characterise the weighting subspaces associated with two approximation techniques for solving ordinary differential equations: the Tau Method [E.L. Ortiz, The Tau Method, SIAM J. Numer. Anal. 6 (1969) 480-92] and the orthogonal collocation method. We show that approximations constructed by means of these two methods are always expressible in terms of a prescribed orthogonal polynomials basis, by projection on a suitably chosen finite dimensional weighting subspace. We show, in particular, that collocation is a special Tau Method with a twisted basis.     

Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.     

A Tau Method with Perturbed Boundary Conditions for Certain Ordinary Differential Equations

Ortiz recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) a perturbed version of the given differential equation, and (ii) the imposed supplementary conditions exactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions are perturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proportional to a Chebyshev polynomial.     

An error analysis of the tau method for a class of singularly perturbed problems for differential equations

By using classical results of Poincaré and Birkhoff we discuss the existence and uniqueness of solution for a class of singularly perturbed problems for differential equations. The Tau method formulation of Ortiz [6] is applied to the construction of approximate solutions of these problems. Sharp error bounds are deduced. These error bounds are applied to the discussions of a model problem, a simple one-dimensional analogue of Navier-Stokes equation, which has been considered recently by several authors (see [2], [3], [8], [10]). Numerical results for this problem [8] show that the Tau method leads to more accurate approximations than specially designed finite difference or finite element schemes.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

Improving the Accuracy of Chebyshev Tau Method for Nonlinear Differential Problems

The spectral properties convergence of the Tau method allow to obtain good approximate solutions for linear differential problems advantageously. However, for nonlinear differential problems the method may produce ill-conditioned matrices issued from the approximations obtained in the iterations from the linearization process. In this work we introduce a procedure to approximate nonlinear terms in the differential equations and a new way to build the corresponding algebraic problem improving the stability of the overall algorithm. Introducing the linearization coefficients of orthogonal polynomials in the Tau method within the iterative process, we can go further in the degree to approximate the solution of the differential problems, avoiding the consequences of ill-conditioning.     

A note on accurate estimations of the local truncation error of polynomial methods for differential equations

We give sharp error estimations for the local truncation error of polynomial methods for the approximate solution of initial value problems. Our analysis is developed for second order differential equations with polynomial coefficients using the Tau Method as an analytical tool. Our estimates apply to approximations derived with the latter, with collocation and with techniques based on series expansions. An example on collocation is given, to illustrate the use of our estimates.     

A matrix formulation of the Tau Method for Fredholm and Volterra linear integro-differential equations

In this paper we obtain the matrix Tau Method representation of a general boundary value problem for Fredholm and Volterra integro-differential equations of orderv. Some theoretical results are given that simplify the application of the Tau Method. The application of the Tau Method to the numerical solution of such problems is shown. Numerical results and details of the algorithm confirm the high accuracy and userfriendly structure of this numerical approach.     

The Buchstab’s function and the operational Tau Method

In this article we discuss some aspects of operational Tau Method on delay differential equations and then we apply this method on the differential delay equation defined byw(u) = 1/u for 1 ≤u ≤ 2 and(uw(u))′ = w(u-1) foru ≥ 2, which was introduced by Buchstab. As Khajah et al.[l] applied the Recursive Tau Method on this problem, they had to apply that Method under theMathematica software to get reasonable accuracy. We present very good results obtained just by applying the Operational Tau Method using a Fortran code. The results show that we can obtain as much accuracy as is allowed by the Fortran compiler and the machine-limitations. The easy applications and reported results concerning the Operational Tau are again confirming the numerical capabilities of this Method to handle problems in different applications.     

The adaptive operational Tau method for systems of ODEs

As the Tau method, like many other numerical methods, has the limitation of using a fixed step size with some high degree (order) of approximation for solving initial value problems over long intervals, we introduce here the adaptive operational Tau method. This limitation is very much problem dependent and in such case the fixed step size application of the Tau method loses the true track of the solution. But when we apply this new adaptive method the true solution is recovered with a reasonable number of steps. To illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations (ODEs). The numerical results confirm the efficiency of the method.     

A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method

In this paper, we propose a Tau method for solving the singular Lane–Emden equation—a nonlinear ordinary differential equation on a semi‐infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane–Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well‐known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd.     

Approximating Rough Stochastic PDEs

We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.     

Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations

A method is presented for obtaining recurrence relations for the coefficients in ultraspherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zx), -1\leq x\leq 1,z\in C,|z|\leq 1$, where y is a solution of a linear differential equation. In particular, rational approximations of $y(z)$ result if $x$ is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.     

非线性积分微分方程有限元逼近的L_∞-模误差界

在气体扩散、热传导等众多物理问题中,经常出现如下非线性抛物型积分-微分方程:     

The L<Superscript>2</Superscript>-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations

The operational Tau method, a well known method for solving functional equations is employed to approximate the solution of nonlinear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. The unique solvability of the linear Tau algebraic system is discussed. In addition, we provide a rigorous convergence analysis for the Legendre Tau method which indicate that the proposed method converges exponentially provided that the data in the given FIDE are smooth. To do so, Sobolev inequality with some properties of Banach algebras are considered. Some numerical results are given to clarify the efficiency of the method.     

SM-stability of operator-difference schemes

The spectral mimetic (SM) properties of operator-difference schemes for solving the Cauchy problem for first-order evolutionary equations concern the time evolution of individual harmonics of the solution. Keeping track of the spectral characteristics makes it possible to select more appropriate approximations with respect to time. Among two-level implicit schemes of improved accuracy based on Padé approximations, SM-stability holds for schemes based on polynomial approximations if the operator in an evolutionary equation is self-adjoint and for symmetric schemes if the operator is skew-symmetric. In this paper, additive schemes (also called splitting schemes) are constructed for evolutionary equations with general operators. These schemes are based on the extraction of the self-adjoint and skew-symmetric components of the corresponding operator.     

On Numerical Approximations of Fractional and Nonlocal Mean Field Games

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

     

Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval

Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Received October 6, 1997 / Revised version received July 22, 1999 / Published online June 21, 2000     

Numerical solution of transient heat conduction equation with variable thermophysical properties by the Tau method

In this article, we proposed the operational approach to the Tau method for solving linear and nonlinear one‐dimensional transient heat conduction equations with variable thermophysical properties which can involve heat generation term. To solve heat conduction equation, first we recall the Tau method to obtain a matrix form of the governing differential equation. Then boundary and initial conditions are transformed into a matrix form. Finally the resulting systems of linear or nonlinear algebraic equations are given. Afterwards, efficient error estimation is also introduced for this method. Some numerical examples are given to illustrate the efficiency and high accuracy of the proposed method and also results are compared with solutions obtained by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 964–977, 2014