A piecewise constant argument method for solving nonlinear differential equations of Lane–Emden type

In this note, a novel piecewise constant argument method (PCAM) is proposed for the numerical solution of singular Lane–Emden-type differential equations subject to initial value problems. To approximate the original problem, a family of differential equations with piecewise constant arguments (DEPCA) is constructed, depending on a positive integer n, where n represents the number of subintervals. The existence and uniqueness of the solution to the resulting second-order singular DEPCA are established. Furthermore, it is proven that the PCAM solution converges to the exact solution as n→∞. Several test problems arising in physical applications are solved to illustrate the applicability of the method. The numerical experiments demonstrate that PCAM is both effective and highly accurate. Comparative results indicate that the proposed method outperforms existing approaches, including artificial neural networks (ANN), the homotopy perturbation method (HPM) and the Laplace Decomposition Method (LDM).

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