Nonlinear Chebyshev fitting from the solution of ordinary differential equations

The nonlinear Chebyshev approximation of real-valued data is considered where the approximating functions are generated from the solution of parameter dependent initial value problems in ordinary differential equations. A theory for this process applied to the approximation of continuous functions on a continuum is developed by the authors in [17]. This is briefly described and extended to approximation on a discrete set. A much simplified proof of the local Haar condition is given. Some algorithmic details are described along with numerical examples of best approximations computed by the Exchange algorithm and a Gauss-Newton type method.     

Fixed mesh approximation of ordinary differential equations with impulses

When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite. So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates. Received March 15, 1996 / Revised version received January 3, 1997     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

Convergence rates for adaptive approximation of ordinary differential equations

Summary This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by . The alternative version with constant tolerance stops with total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L ¹ -norm of the error density. Mathematics Subject Classification (2000):65Y20, 65L50, 65L70This work has been supported by the EU–TMR project HCL # ERBFMRXCT960033, the EU–TMR grant # ERBFMRX-CT98-0234 (Viscosity Solutions and their Applications), the Swedish Science Foundation, UdelaR and UdeM in Uruguay, the Swedish Network for Applied Mathematics, the Parallel and Scientific Computing Institute (PSCI) and the Swedish National Board for Industrial and Technical Development (NUTEK).     

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.     

Best approximation estimates resulting from the Chebyshev criterion

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 148–150, 1991.     

Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations

The aim of this article is to present an analytical approximation solution for linear and nonlinear multi-order fractional differential equations (FDEs) by extending the application of the shifted Chebyshev operational matrix. For this purpose, we convert FDE into a counterpart system and then using proposed method to solve the resultant system. Our results in solving four different linear and nonlinear FDE, confirm the accuracy of proposed method.     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

A variational principle for adaptive approximation of ordinary differential equations

Summary A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error=local errorweight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms. Mathematics Subject Classification (2000):65L70, 65G50This work has been supported by the EU–TMR project HCL # ERBFMRXCT960033, the EU–TMR grant # ERBFMRX-CT98-0234 (Viscosity Solutions and their Applications), the Swedish Science Foundation, UdelaR and UdeM in Uruguay, the Swedish Network for Applied Mathematics, the Parallel and Scientific Computing Institute (PSCI) and the Swedish National Board for Industrial and Technical Development (NUTEK).     

The use of Chebyshev series for approximate analytic solution of ordinary differential equations

Application of Chebyshev series to solve ordinary differential equations is described. This approach is based on the approximation of the solution to a given Cauchy problem and its derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas. It is shown that the proposed approach can be applied to formulate an approximate analytical method for solving Cauchy problems. A number of examples are considered to illustrate the obtaining of approximate analytical solutions in the form of partial sums of shifted Chebyshev series.     

Best approximation and numerical methods in solution of differential equations

Let Q(x) bee polynomial of degree q interpolating x^m at the points $\text{x}{}_{\mathrm{<mtext>i</mtext>}}\text{,}\text{i}\text{= 0, 1, /3.,}\text{q}$, where $\text{x}{}_{\mathrm{<mtext>i</mtext>}}$ are zeros of the Tchebysheff polynomial of degree q + 1 on the interval [0, 1]. If q is of order √m, then Q(x) approximates x^m well enough. This result is used to obtain a good approximation to the solution of a system of linear differential equations.     

Existence and approximation of solutions to nonlinear hybrid ordinary differential equations

In this note, we consider the initial value problem for first order nonlinear hybrid ordinary differential equations and we discuss the existence and approximation of the solutions. The main results are related to a recent work by Dhage et al. (2014) through the restructuring of some of the hypotheses imposed and the extension of some of the results there. In addition, we provide an example to illustrate the applicability of the abstract results developed.     

Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations

Two-parameter families of predictor-corrector methods based upon a combination of Adams- and Nyström formulae have been developed. The combinations use correctors of order one higher than that of the predictors. The methods are chosen to give optimal stability properties with respect to a requirement on the form and size of the regions of absolute stability. The optimal methods are listed and their regions of absolute stability are presented. The efficiency of the methods is compared to that of the corresponding Adams methods through numerical results from a variable order, variable stepsize program package.     

Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011