Orthogonal functions and polynomials have been used by many authors for solving various problems. The main idea of using orthogonal basis is that a problem reduces to solving a system of linear or nonlinear algebraic equations by truncated series of orthogonal basis functions for solution of problem and using the operational matrices. Here we use Legendre wavelets basis on interval [0, 1]. Some...

A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel'fand-Levitan type, the classic example being the Korteweg-de Vries equation. An ordinary differential equation is of Painlev type if the only singularities of its solutions in the complex plane are poles. It is shown that, under cer...

This paper presents the solution to the following optimization problem: What is the shape of the region that minimizes the average Lp distance between all pairs of points if the area of this region is held fixed? (The Lp-distance between two points x = (x1, x2) and y = (y1, y2) in < 2 is (|x1 − y1| p + |x2 − y2| p).) We use variational methods to determine the shape of the region that minimizes...