Abstract. The main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. Properties of the Legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. Afterwards, an iterative algorithm is examined for solvin...

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagn...

In this report, we present two mathematical results which can be useful in a variety of settings. First, we present an analysis of Legendre polynomials triple product integral. Such integrals arise whenever two functions are multiplied, with both the operands and the result represented in the Legendre polynomial basis. We derive a recurrence relation to calculate these integrals analytically. W...

This paper presents a new set of hemispherical basis functions dedicated to hemispherical data representation. These functions are derived from associated Legendre polynomials. We demonstrate the usefulness of this basis for representation of surface reflectance functions, rendering using environment maps and for efficient global illumination computation using radiance caching. We show that our...

We extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkinmethod while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse. The algorithm can be implemented efficiently and i...

In this paper, a numerical method is proposed for solving optimal control problem of Volterra integral equations using radial basis functions (RBFs) for approximating unknown function. Actually, the method is based on interpolation by radial basis functions including multiquadrics (MQs), to determine the control vector and the corresponding state vector in linear dynamic system while minimizing...

A Legendre spectral Galerkin method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of two Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplaci...

This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to hp-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The...