In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled diierential equations, exponentials of vector elds, and Lie brackets. We show by means of an example |the computation of control variations that give rise to the Legendre-Clebsch condition| how a good choice of formalism , based on expanding diieomorphisms as products of expon...

Pseudo Random Noise (PRN) codes are essential part in space communication. A pseudo random noise binary sequence is a semi-random sequence in the sense that it appears random within the sequence length, fulfilling the needs of randomness. The objective of the paper is to generate different types of PN sequences i.e. Legendre sequence, Weil sequence, Sidel’nikov sequence and polyphase Sidel’niko...

We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The is applied to time discretization and the term dealt with method. scheme results in simple algebraic system by choosing appropriate basis functions. Optimal error estimates $ L^2 $-norm single multiple interval method...

The spherical harmonics Y l m form a basis of the vector space V (of dimensions 2l + 1) of the eigen functions of the Laplacian on the sphere, with eigen value λl = −l (l+ 1). Here we show the existence of an other basis Φj for V, where Φj(X) ≡ (X · Nj), the l power of the scalar product of the current point with a specific null vector Nj . This very simple formulation makes calculations very e...

We devise a framework encompassing the classical theory of characteristics and the theory valid in the convex case recently obtained by R.T. Rockafellar and P. Wolenski. It relies on a notion of transform introduced by I. Ekeland. It involves a class of functions called Ekeland functions which is large enough to encompass convex functions, concave functions and linear-quadratic functions, as we...

This paper presents a simple matrix form for degree elevation of interval Bezier curve using LegendreBernstein basis transformations. The four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original interval Bezier curve are obtained. These four fixed Bezier curves are expressed in terms of the Legendre polynomials. The process of degree elevations r times are app...

Abstract This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order version of Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family basis functions, are employed to generate operational matrix method this A new related these polynomials is extracted and construct presented method. Using approach, algebr...

The Legendre-Stirling numbers are the coeffi cients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejoh...