Two related approximation problems are formulated and solved in Hardy spaces of the disc and annulus. With practical applications in mind, truncated versions of these problems are analysed, where the solutions are chosen to lie in finite-dimensional spaces of polynomials or rational functions, and are expressed in terms of truncated Toeplitz operators. The results are illustrated by numerical e...

This paper considers a progressive solitary wave of permanent form in an ideal fluid of constant depth and explores Davies’ approximation [Proc. R. Soc. Lond. A, 208 (1951), pp. 475– 486] with high-order corrections to Levi-Civita’s surface condition for the logarithmic hodograph variable. Using a complex plane that was originally introduced by Packham [Proc. R. Soc. Lond. A, 213 (1952), pp. 23...

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and reliable. Therefore, modified numer...

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary lay...

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and disp...

This paper is an application of the ideas of the Born-Oppenheimer (or slow/fast) approximation in molecular physics and of the Isaacson (or shortwave) approximation in classical gravity to the canonical quantization of a perturbed minisuperspace model of the kind examined by Halliwell and Hawking. Its aim is the clarification of the role of the semiclassical approximation and the backreaction i...

We generalize the notion of an approximation space introduced in 8]. In tolerance approximation spaces we deene the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems. The tolerance relation deened by the so called uncertainty function or the positive region of a given partition of o...

For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg– de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of small anharmonicity. We prove control of the approximation error for the travelling solutions satisfying differential advance-delay equations, as well as control of the approxi...

We present approximation algorithms for cutting out polygons with line cuts and ray cuts. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possib...

In this paper, a reliable approach is introduced to approximate periodic solutions of a system of coupled integrable dispersionless. The system is firstly, transformed into an ordinary differential equation by wave transformation. The solution of ODE is obtained by the homotopy perturbation method. To show the periodic behavior of the solution, a modification based on the Laplace transforms and...