A sound pulse is scattered by a sphere leading to an initial–boundary value problem for the wave equation. A method for solving this problem is developed using integral representations involving Legendre polynomials in a similarity variable and Volterra integral equations. The method is compared and contrasted with the classical method, which uses Laplace transforms in time combined with separa...

Most of the reconstruction-based robust adaptive beamforming (RAB) algorithms require covariance matrix reconstruction (CMR) by high-complexity integral computation. A Gauss-Legendre quadrature (GLQ) method with highest algebraic precision in interpolation-type is proposed to reduce complexity. The interference angular sector RAB regarded as GLQ range, and zeros three-order Legendre orthogonal ...

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. We develop a twovariable refinement WD(s,t) of the Jones polynomial that is invariant under symmetric Reidemeister moves. If D is a symmetric union diagram, representing a ribbon knot K, then the polynomial WD(s,t) nicely reflects their topological propertie...

In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...

We consider the sequences over the finite field of four elements obtained by inverse Gray mapping from a pair of binary sequences. We derive the linear complexity and the minimal polynomial of sequences constructed from Legendre sequences, Hall’s sextic sequences and twin-prime sequences using the technique proposed by Tang, Ding, Lim, Kim et al.

The main aim of this article is to generalize the Legendre operational matrix to the fractional derivatives and implemented it to solve the nonlinear multi-order fractional differential equations. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used. The main characteristic behind the approach using this technique is that...

A conducting spherical shell with a circular orifice of half angle θ0 is at electric potential V0. Show that the difference between the charge densities on the inner and outer surfaces is independent of position, and estimate the ratio of the electric charge on the inner surface to that on the outer. Correct results can be inferred from “elementary” arguments based on superposition, and more “e...

The three-dimensional high-order simulation algorithm HOSIM is developed to simulate complex nonlinear and non-Gaussian systems. HOSIM is an alternative to the current MP approaches and it is based upon new high-order spatial connectivity measures, termed high-order spatial cumulants. The HOSIM algorithm implements a sequential simulation process, where local conditional distributions are gener...

in this paper, we use the continuous legendre wavelets on the interval [0,1] constructed by razzaghi m. and yousefi s. [6] to solve the linear second kind integral equations. we use quadrature formula for the calculation of the products of any functions, which are required in the approximation for the integral equations. then we reduced the integral equation to the solution of linear algebraic ...