In this paper, numerical solutions of singular initial value problems are obtained by the Haar wavelet collocation method (HWCM). The HWCM is a numerical method for solving integral equations, ordinary and partial differential equations. To show the efficiency of the HWCM, some examples are presented. This method provides a fast convergent series of easily computable components. The errors of HWCM are also computed. Through this analysis, the solution is found on the coarse grid points and then converging toward higher accuracy by increasing the level of the Haar wavelet. Comparisons with exact and existing numerical methods (adomian decomposition method (ADM) & variational iteration method (VIM)) solutions show that the HWCM is a powerful numerical method for the solution of the linear and non-linear singular initial value problems. The Haar wavelet adaptive grid method (HWAGM) based solutions show the excellent performance for the proposed problems. Ó 2015 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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For the classical risk process R(t) that is linear increasing with slope 1 between downward jumps of i.i.d. random sizes at the points of a homogeneous Poisson process we consider the level-crossing process C(x) = (L(x), (Ai (x), Bi (x))1≤ i ≤ L ( x )), where L(x) is the number of jumps from (x, ∞) to (−∞, x] and Ai (x) (Bi (x)) are the distances from x to R(t) after (before) the ith jump of this kind. It is shown that if R(·) has a drift toward infinity, C(·) is a stationary Markov process; its transition probabilities are determined. As an application we derive the expected value E(L(x)L(x + y)).