An efficient two step Laplace decomposition algorithm for singular Volterra integral equations

Singular integral equation that has enormous applications in applied problems including fluid mechanics, biomechanics, electromagnetic theory and chemistry applications such as heat conduction, crystal growth and electrochemistry. An integral equation is called a singular integral equation if one or both limits of integration become infinite, or if the kernel of the equation becomes infinite at one or more points in the interval of integration. Norwegian mathematician Niels Abel who invented them in 1823, in his research of mathematical physics (Jerri, 1999; Rahman, 2007). There are many numerical and analytical schemes such as finite element method, finite difference method and perturbation methods can be used to obtain an approximate solution for the model problem. However, there exist many difficulties such as a mesh refinement, a stability condition and selection of small and large parameters, etc. To avoid these difficulties, decomposition method was introduced (Adomian, 1994; Jafari and Gejji, 2006a, b, c) which is a very powerful method for solving linear and non-linear problems in many fields. Recently, a modification of Laplace decomposition algorithm (LDA) was proposed (Majid et al., 2011; Hussain and Khan, 2010). The modified decomposition algorithm needs only a slight variation from the standard LDA and has been shown to be computationally efficient. The modified LDA (MLDA) was