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Abstract: In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C 2-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented. [Copyright &y& Elsevier]

This article aims to present a new approach based on C 1 -cubic splines introduced by Sallam and Naim Anwar [Sallam, S. and Naim Anwar, M. (2000). Stabilized cubic C 1 -spline collocation method for solving first-order ordinary initial value problems, Int. J. Comput. Math. , 74 , 87-96.], which is A-stable, for the time integration of parabolic equations (diffusion or heat equation). The introduced method is an example of the so-called method of lines (the solution is thought to consist of space discretization and time integration), which is an extension of the 1/3-Simpson's finite-difference scheme. Our main objective is to prove the unconditional stability of the proposed method as well as to show that the method is convergent and is of order O ( h 2 ) + O ( k 4 ) i.e. it is a fourth-order in time and second-order in space. Computational results also show that the method is relevant for long time interval problems. [ABSTRACT FROM AUTHOR]

A new procedure based on sixth degree (Hexic) C 2 -Spline for the numerical integration of the second order initial value problems (IVPs) y^{\prime\prime}=f(x,y) , including those possessing oscillatory solutions, is presented. The proposed method is essentially an implicit sixth order one-step method. Stability analysis shows that the method possesses (0, 75.3)\bigcup (130.2, 201.9) as interval of periodicity and/or absolute stability. In addition, the method has phase-lag (dispersion) of order six with actual phase-lag H^{6}/774144 . Convergence results yield error bounds \parallel\! s^{(r)}-y^{(r)}\!\parallel\,=O\left(h^{6}\right),r=0,1 , in the uniform norm, provided y\in C^{8}[0,b] . Furthermore, it turns out that the method is a continuous extension of a sixth order four-stage Runge-Kutta (-Nyström) method. Numerical experiments will also be considered. [ABSTRACT FROM AUTHOR]