Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function

Jyotirmoy Mouley,B N Mandal,M M Panja

doi:10.1007/s40096-019-00312-w

Abstract

This paper is concerned with obtaining approximate numerical solution of a classical integral equation of some special type arising in the problem of cruciform crack. This integral equation has been solved earlier by various methods in the literature. Here, approximation in terms of Daubechies scale function is employed. The numerical results for stress intensity factor obtained by this method for a specific forcing term are compared to those obtained by various methods available in the literature, and the present method appears to be quite accurate.

Highlights

Integral equations occur naturally in many areas of mathematical physics
Many engineering and applied science problems arising in water waves, potential theory and electrostatics are reduced to solving integral equations
This is an integral equation of some special type since the kernel L(x, t) has singularity at (0, 0) only. f(x) is a prescribed function relating to the internal pressure given by

Summary

Introduction

Integral equations occur naturally in many areas of mathematical physics. Many engineering and applied science problems arising in water waves, potential theory and electrostatics are reduced to solving integral equations. Tang and Li [9] solved the integral equation approximately by employing Taylor series expansion for the unknown function and obtained very accurate numerical estimates for the stress. Expansion in terms of Bernstein polynomials or Haar functions or Legendre multi-wavelets suggest expansion in terms of other functions such as Daubechies scale functions since these provide a somewhat new tool in the numerical solution of integral equations. The knowledge of the low-pass filter coefficients in two-scale relation is required throughout the calculation For these reasons, Daubechies scale function is used as an efficient and new mathematical tool to solve integral equations. The integral equation (1.1) produces a system of linear equations in the unknown coefficients After solving this linear system, the unknown function (x) is evaluated at x = 1 so as to obtain numerically the value of the stress intensity factor. By Isn , we mean the interior scale function whose supports are contained in [a, b]

Method of solution

Conclusion