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.

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Fractional integro-differential equation (FIDE) with weakly singular kernel is an important topic in mathematics and engineering dealing with mathematical modeling and simulation of numerous systems and processes. A wavelet-based collocation technique has been developed here to obtain approximate numerical solution of a FIDE with weakly singular kernel. The present method avoids complicated integrations and elaborate numerical calculations. The multiscale error approximation associated with this method has also been explained. The efficiency of the proposed method has been demonstrated by including some illustrative examples. K E Y W O R D Scollocation method, fractional derivative, multiscale basis of Daubechies family, weakly singular fractional integro-differential equation Here D y(x) represents the fractional derivative of order of the unknown function y(x) and the fractional derivative is understood in Caputo sense in this paper. The explicit forms of V(x, t) and f (x) are known and they are considered sufficiently smooth to confirm the existence and uniqueness of solution. 4