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

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## 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.

## Keywords

Cruciform crack Integral equation Daubechies scale function Daubechies wavelet Gauss–Daubechies quadrature rule## Introduction

*f*(

*x*) is a prescribed function relating to the internal pressure given by

*f*(

*x*). 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 intensity factor. They made use of Cramer’s rule in the mathematical analysis so that if one increases the number of terms in the approximation the calculation becomes unwieldy so as to make the method unattractive. Bhattacharya and Mandal [1] solved the integral equation approximately by two different methods, one is based on expansion of the unknown function in terms of Bernstein polynomials and the other is based on expansion in terms of rationalized Haar functions. Singh and Mandal [7] also solved it by using Legendre multi-wavelets. All these methods provide numerical results for \(\chi (1)\) which are very close to exact results given by Stallybrass [8]. 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.

In this paper, Daubechies scale functions are employed to expand the unknown function \(\chi (x)\). *K*-Daubechies scale function is employed to find approximate solution of integral equation taking \(K=3\). It may be noted that \(K=1\) corresponds to Haar wavelets. As the result can be improved taking larger value of *K*, so the results obtained by using *K*-Daubechies scale function are better than the results using the rationalized Haar functions. Though Legendre multi-wavelets give satisfactory results, *K*-Daubechies scale function has some interesting features like compact support, fractal nature and no explicit form at all resolutions. Only 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. At \(x=1\), the expansion of \(\chi (x)\) reduces to a finite expansion because most of Daubechies scale functions vanish. Actually, the integral equation (1.1) produces a system of linear equations in the unknown coefficients. After solving this linear system, the unknown function \(\chi (x)\) is evaluated at \(x=1\) so as to obtain numerically the value of the stress intensity factor. For different values of \(\sigma\) in the expression of internal pressure *f*(*x*) given by (1.3), \(\chi (1)\) is obtained and compared to known results available in the literature. It is found that the method is quite accurate as the approximate values of \(\chi (1)\) obtained by the present method are seen to differ negligibly from exact values.

## Basic properties of Daubechies scale function and wavelets

*K*-Daubechies scale function \((K\ge 1)\) has 2

*K*scaling coefficients and has compact support \([0, 2K-1]\). It may be noted that \(K=1\) corresponds to the Haar wavelets. The two-scale relations of scale functions and wavelet functions are given by

*T*and the scale transformation D, defined as

*K*-Daubechies scale function are determined using conditions (2.6), (2.7) and (2.8) given below,

*a*and

*b*are integers. In order to apply the machinery of

*K*-Daubechies scale function on a finite interval [

*a*,

*b*] , we have to modify the properties of scale function \(\phi _{sn}^{L ~\text {or}~ R}\) for \(2^{s}a-(2K-2)\le n \le 2^{s}a-1\) and \(2^{s}b-(2K-2)\le n\le 2^{s}b-1\), whose supports overlap partially with the finite interval [

*a*,

*b*]. The superscripts

*L*and

*R*stand for left and right partial overlaps with [

*a*,

*b*], respectively. By \(\phi _{sn}^{I}\), we mean the interior scale function whose supports are contained in [

*a*,

*b*].

## Method of solution

*a*,

*b*] is [0, 1]. To find the approximate solution of (1.1), we approximate the unknown function \(\chi \left( x\right)\) defined in [0, 1] in terms of Daubechies scale functions in the form

*s*.

*m*, Eq. (3.2) is reduced to the form

*m*and

*n*depends on scales

*s*and

*K*. As the support of \(\phi \left( x\right)\) is \(\left[ 0,2K-1\right]\) and \(K(\ge 1)\) is a positive integer, so \(m, n \in \left[ -(2K-2),2^{s}-1\right]\).

*n*in (3.7) satisfy the range \(1-2K+2^{s}\le n\le 2^{s}\). If we take \(( K=3)\)-Daubechies scale function, we need values of \(\phi \left( p\right)\) for \(p = 5, 4, 3, 2, 1, 0\). The detailed tricks for calculation of (3.4), (3.5) and (3.6) are described below.

*f*(

*x*) in (1.3) and the two-scale relation (2.1), the expression in (3.4) reduces to the form

*m*or \(n \ge 2^{s}\) or \(\mid m-n \mid \ge 2K-1\) gives all the values of \(N_{mn}^{L~\text {or}~R}\) for \(m,n= -(2K-2), -(2K-3), \ldots ,-1\) and \(2^{s}-(2K-2), 2^{s}-(2K-3), \ldots ,2^{s}-1\). Here, the superscripts

*L*and

*R*stand for left edge and right edge of [0, 1], respectively. Again if \(0\le m,n \le 2^{s}-(2K-1)\), we mean \(N_{mn}\) by \(N_{mn}= N_{mn}^{I}=\delta _{mn}\). For \(( K=3)\)-Daubechies scale function, the numerical values of \(N_{mn}^{L ~\text {or}~ R}\) and the corresponding values of the inverse of the matrix formed by the elements \(N_{mn}^{L~\text {or}~ R}\) are tabulated in different tables by Panja and Mandal [4].

*m*,

*n*, the values of \(I_{m,n}^{s}\) cannot be determined using the relation (3.18).

*m*or \(n \le -(2K-1)\) or

*m*or \(n \ge 2^{s}\). We present here the numerical values of \(I_{m,n}^{s}\) for( \(K=3\))-Daubechies scale functions taking \(s =3\) for those values of

*m*and

*n*for which \(\Theta _{m,n,l_{1},l_{2}}\left( x,t\right)\) has singularity at (0, 0). Table 1 shows the values of \(I_{m,n}^{s}\) for \(N=5,\) whereas Table 2 shows the values of \(I_{m,n}^{s}\) for \(N=7\).

Numerical values of \(I_{m,n}^{s}\) for \(K=3, N=5\)

\(I_{m,n}^{s}\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | 0 |
---|---|---|---|---|---|

\(-4\) | \(6.20189\times {10^{-7}}\) | \(6.17887\times 10^{-6}\) | \(-\)\(2.42358\times {10^{-4}}\) | \(1.32994\times {10^{-3}}\) | \(~8.65630\times {10^{-4}}\) |

\(-3\) | \(1.75110\times {10^{-5}}\) | \(5.53604\times {10^{-4}}\) | \(-\)\(1.32760\times {10^{-3}}\) | \(~8.53732\times {10^{-3}}\) | \(~7.54718\times {10^{-2}}\) |

\(-2\) | \(-\)\(7.09342\times {10^{-5}}\) | \(-\)\(1.88971\times {10^{-3}}\) | \(-\)\(8.75770\times {10^{-3}}\) | \(-\)\(4.55777\times {10^{-2}}\) | \(-\)\(1.24792\times {10^{-2}}\) |

\(-1\) | \(3.23710\times {10^{-4}}\) | \(1.26285\times {10^{-2}}\) | \(-\)\(7.43785\times {10^{-2}}\) | \(2.79172\times {10^{-1}}\) | \(1.64992\times {10^{-1}}\) |

0 | \(9.93316\times 10^{-7}\) | \(4.07921\times {10^{-4}}\) | \(-\)\(6.06907\times {10^{-3}}\) | \(4.73939\times {10^{-2}}\) | \(3.27281\times {10^{-1}}\) |

Numerical values of \(I_{m,n}^{s}\) for \(K=3, N=7\)

\(I_{m,n}^{s}\) | \(- 4\) | \(-3\) | \(-2\) | \(-1\) | 0 |
---|---|---|---|---|---|

\(-4\) | \(6.20146\times {10^{-7}}\) | \(6.18006\times 10^{-6}\) | \(-\)\(2.42335\times {10^{-4}}\) | \(1.32979\times {10^{-3}}\) | \(~8.65506\times {10^{-4}}\) |

\(-3\) | \(1.75097\times {10^{-5}}\) | \(5.53569\times {10^{-4}}\) | \(-\)\(1.32778\times {10^{-3}}\) | \(~8.53718\times {10^{-3}}\) | \(~7.54618\times {10^{-2}}\) |

\(-2\) | \(-\)\(7.09290\times {10^{-5}}\) | \(-\)\(1.88961\times {10^{-3}}\) | \(-\)\(8.75638\times {10^{-3}}\) | \(-\)\(4.55772\times {10^{-2}}\) | \(-\)\(1.24797\times {10^{-2}}\) |

\(-1\) | \(3.23683\times {10^{-4}}\) | \(1.26275\times {10^{-2}}\) | \(-\)\(7.43729\times {10^{-2}}\) | \(2.79151\times {10^{-1}}\) | \(1.64981\times {10^{-1}}\) |

0 | \(9.91086\times 10^{-7}\) | \(4.07828\times {10^{-4}}\) | \(-\)\(6.06822\times {10^{-3}}\) | \(4.73891\times {10^{-2}}\) | \(3.27262\times {10^{-1}}\) |

## Numerical results

Approximate values of \(\chi \left( 1\right)\) for different \(\sigma\)

\(\sigma\) | Exact | \(K=3, N=5\) | Relative error | \(K=3, N=7\) | Relative error |
---|---|---|---|---|---|

1 | 0.86354 | 0.863656 | 0.000134 | 0.863660 | 0.000139 |

2 | 0.57547 | 0.575551 | 0.000141 | 0.575463 | 0.000012 |

3 | 0.46350 | 0.463562 | 0.000134 | 0.463424 | 0.000164 |

4 | 0.39961 | 0.399170 | 0.001101 | 0.398996 | 0.001536 |

5 | 0.35681 | 0.355325 | 0.004162 | 0.355123 | 0.004728 |

6 | 0.32549 | 0.322496 | 0.009198 | 0.322270 | 0.009893 |

7 | 0.30125 | 0.296382 | 0.016159 | 0.296137 | 0.016973 |

8 | 0.28176 | 0.274740 | 0.025057 | 0.274476 | 0.025852 |

9 | 0.26564 | 0.256272 | 0.035266 | 0.255994 | 0.036312 |

10 | 0.25201 | 0.240179 | 0.046946 | 0.239886 | 0.048109 |

The table shows the exact values of \(\chi (1)\)\((\sigma =1, 2, \ldots ,10)\) according to Stallybrass [8] and the results obtained by the present method with their relative errors.

## Conclusion

Here, a numerical scheme based on expansion in terms of *K*-Daubechies scale function is employed for obtaining approximate numerical estimates of an integral equation of some special type arising in the classical problem of cruciform crack in elasticity. Comparison between the numerical results obtained by the present method with the exact results obtained by Stallybrass [8] shows that the method is quite accurate. The method works nicely for moderate values of *K* (e.g., \(K=3\)). The results can be further improved taking larger values of \(K ~(K>3\)).

## Notes

### Acknowledgements

The authors thank the Reviewer and the Associate Editor for comments and suggestions to revise the paper in the present form. JM acknowledges financial support from University Grants Commission, New Delhi, for the award of research fellowship (File No. 16-9(june2017/2018(NET/CSIR))).

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