In an aplanatic optical system, where the entrance pupil is not located at the front focal plane, a corresponding quadratic phase is expected at the back focal plane. However, tightly focused optical fields evaluated by methods based on the traditional Richards-Wolf approach, lacks this quadratic phase. In the current work we calculate the focused field, for both high and medium numerical apert...

We show how Mathematica can be used to obtain numerical solutions of integral equations by exploiting a combination of iteration and interpolation. The efficacy of the method is demonstrated by considering three classical integral equations of applied mathematics: Love’s equation for the condenser problem, Theodorsen’s equation associated with conformal mapping, and Nekrasov’s equation arising ...

We study the thermoelastic deformation of an elastic layer. The upper surface of the medium is subjected to a uniform thermal field along a circular area while the layer is resting on a rigid smooth circular base. The doubly mixed boundary value problem is reduced to a pair of systems of dual integral equations. The both system of the heat conduction and the mechanical problems are calculated b...

the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. th...

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 ...

We present a novel computational methodology based on Nyström discretizations to produce fast and very accurate solutions of acoustic and electromagnetic problems in small numbers of Krylov-subspace iterative solvers. At the heart of our approach are integral equation formulations that exhibit excellent spectral properties. In the case of scattering from perfectly conducting structures, and jus...

In this article the problem of an interface fracture between two isotropic linear elastic materials is studied within a continuum modeling framework, which incorporates important nanoscale effects. The proposed model of bi-material crack ascribes curvature-dependent surface tension to both the fracture surfaces and the solid-solid interface. Further, it uses as boundary conditions the jump mome...

Fuzzy integral equations have a major role in the mathematics and applications.In this paper, general fuzzy integral equations with nonlinear fuzzykernels are introduced. The existence and uniqueness of their solutions areapproved and an upper bound for them are determined. Finally an algorithmis drawn to show theorems better.

This paper gives an ecient numerical method for solving the nonlinear systemof Volterra-Fredholm integral equations. A Legendre-spectral method based onthe Legendre integration Gauss points and Lagrange interpolation is proposedto convert the nonlinear integral equations to a nonlinear system of equationswhere the solution leads to the values of unknown functions at collocationpoints.