We develop a numerical approach for computing the additive, multiplicative and compressive convolution operations from free probability theory. We utilize the regularity properties of free convolution to identify (pairs of) ‘admissible’ measures whose convolution results in a so-called ‘invertible measure’ which is either a smoothly-decaying measure supported on the entire real line (such as th...

This paper deals with the application of fourth kind Chebyshev wavelets (FKCW) in solving numerically a model of HIV infection of CD4+T cells involving Caputo fractional derivative. The present problem is a system of nonlinear fractional differential equations. The goal is to approximate the solution in the form of FKCW truncated series. To do this, an operational matrix of fractional integrati...

This paper describes the behaviour of a flap-type oscillating wave energy converter near a straight coast. The mathematical formulation is based on a linear potential flow model. Application of Green’s theorem to a semi-infinite fluid domain yields a hypersingular integral equation for the velocity potential which is solved using a series expansion of Chebyshev polynomials. Extremes in the hydr...

We consider the β-expansion of 1 which encodes a rational rotation on R/Z under a certain partition. Via transforming its βpolynomial into a linear combination of Chebyshev polynomials, we specify the minimal polynomial of β over the field of rationals. At the end, some number theoretical conjectures are posed with computational evidence to support them.

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval ofintegration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which le...

A mode III crack problem in a functionally graded material (FGM) modeled by anisotropic strain gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths ` and `′, which are responsible for volumetric and surface strain-gradient terms, respectively. The governing differential equation of the problem is derived ass...

A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corr...

This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [18] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains, and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functio...