Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

Abstract

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

On The Studying of Contact Mechanics of a Flat Stamp Graded Coatings Using An Advanced Barycentric Lagrange Interpolation Formula

“…Consider the FGM-coated elastic half-plane with an ordinary stress boundary value problem described in Fig. (1). The structural is a metallic substrate bonded to a metal/ceramic coating with continuous and variable thermo-mechanical properties.…”

“…Which can be calculated by applying the Barycentric Lagrange Interpolation technique [1]. We begin by interpolating the unknown () Pxand data functions () fx by using the advanced single matrix form barycentric interpolate polynomials; each is expressed through four matrices, and one of which is the monomial basis functions matrix.…”

“…For the circular stamp profile, the stress intensity and contact stresses factors are numerically calculated numerically in the form of a singular Fredholm integral equation of the second type. We will investigate in this paper the initial fracture in graded coatings under sliding rigid flat stamp contact loading using the Barycentric Lagrange Interpolation technique [1].…”

“…This problem could be represented in the form of weakly singular Fredholm of the second kind. Shoukralla et al [1], made progress towards the application of some advanced single and double barycentric Lagrange interpolants and how to adapt them to be applicable to completely isolating the kernel singularity and find accurate solutions of the weakly singular Fredholm equations of the second kind which we will use in this paper for studying the effect of many factors such as the coefficient of friction, material inhomogeneity constants, and other length parameters on the critical stresses that may affect the break of the coating.…”

Barycentric Lagrange Interpolation in Matrix-Vector Versions for solving Love’s Integral Equations

“…Most of the aforementioned methods are tedious, make you feel bored in their calculations, and praise the theories of functional analysis, although the solution to these equations has been proven to be convergent, continuous, even, and stable. There are also many different innovative methods and techniques [5][6][7][8][9][10][11][12][13][14][15] for solving Volterra and Fredholm weakly singular integral equations. These methods are suitable for solving Love's integral equations for any parameter, but they are expensive because they contain singularities in the kernels, and sometimes the unknown functions are also singular near the endpoints of the domain of integration.…”