B Wood

On Numerically Solving Nonlinear Volterra Integral Equations with Fewer Computations

A method is described whereby certain nonlinear Volterra integral equations may be numerically solved by computing the solution of a system of ordinary differential equations. In case the kernel is not finitely decomposable, certain two-dimensional approximating techniques are employed. In such a case, there may be a trade-off between computational effort and accuracy. Several explicit error estimates are given, and numerical examples illustrate the applicability of the method as it compares with other methods.

A smoothed projection method for singular, nonlinear Volterra equations

Abstract A modified Galerkin method previously used to approximate the solution of nonlinear Volterra integral equations of the second kind with smooth kernels is generalized to include such equations with singular, monotone kernels of convolution type. Several singular kernel approximations are considered, including positive convolution operators and integral splines. The main results relating to the original integral equation supply error estimates resulting from using the kernel approximations and an approximating system of ordinary differential equations.

A note on solving Volterra integral equations with convolution kernels

A method described previously by the authors is applied to certain smooth kernels of convolution type in order to obtain numerical solutions to certain Volterra integral equations. Several one-dimensional approximation techniques are compared, and the finite Tchebycheff expansion is found to be most effective in terms of decreased computational effort. A number of numerical examples are cited.

Quantitative estimates for Lp approximation with positive linear operators

Abstract Quantitative estimates for approximation with positive linear operators are derived. The results are in the same vein as recent results of Berens and DeVore. Two examples are provided.

Degree of Lp-approximation by certain positive convolution operators

Abstract The degree of L p -approximation for a class of positive convolution operators is investigated. Recent results of De Vore, Bojanic, and Shisha for the uniform approximation by these operators and the K -functional of Peetre are employed to obtain the degree of approximation in terms of the integral modulus of smoothness.

Uniform approximation by linear combinations of Bernstein-type polynomials

Abstract Uniform approximation is considered by linear combinations due to May and Rathore of integral modifications of the Bernstein polynomial introduced by Durrmeyer. The order of uniform approximation is obtained in terms of higher-order modulus of continuity of the function being approximated.

Order of approximation by linear combinations of positive linear operators

Abstract Order of uniform approximation is studied for linear combinations due to May and Rathore of Baskakov-type operators and recent methods of Pethe. The order of approximation is estimated in terms of a higher-order modulus of continuity of the function being approximated.

Unbounded functions and positive linear operators

Abstract The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.

Approximation by discrete operators

Abstract A discrete, positive, weighted algebraic polynomial operator which is based on Gaussian quadrature is constructed. The operator is shown to satisfy the Jackson estimate and an optimal version is obtained.

Local Lp-saturation of positive linear convolution operators

Abstract Local L p -saturation of positive linear convolution operators is investigated. Results are obtained for two important classes of operators previously studied by Bojanic, DeVore, Korovkin and the authors.