Universal Journal of Computational Mathematics Vol. 7(1), pp. 8 - 13
DOI: 10.13189/ujcmj.2019.070102
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On HPM Approximations for the Cumulative Normal Distribution Function


V. K. Shchigolev *
Department of Theoretical Physics, Ulyanovsk State University, Ulyanovsk, 432000, Russian Federation

ABSTRACT

In this paper, some new approximations to the cumulative distribution function of the standard normal distribution via the He's homotopy perturbation method are proposed. There are several methods which provide an approximation of the integral in the formula for the cumulative distribution function by different numerical methods. For the same purpose, we first establish a differential equation of the second order that the cumulative distribution function satisfied subjected with the certain initial conditions. Then we apply the Homotopy Perturbation Method to solve the Cauchy problem for the governing equation. As well known, the result of solving an equation by this method and the convergence rate greatly depend on the choice of homotopy applied. Therefore, we consider two cases in this work. In one case, we construct the homotopy from the idea of simplicity. In the next case, we just follow the procedure of the general approach proposed early. As a result, we obtain several approximations which can be are easily calculated and are better than some other approximations. Numerical comparison shows that our approximations are very accurate.

KEYWORDS
Normal Distribution, Cumulative Distribution Function, Approximations, Homotopy Perturbation Method

Cite this paper
V. K. Shchigolev . "On HPM Approximations for the Cumulative Normal Distribution Function." Universal Journal of Computational Mathematics 7.1 (2019) 8 - 13. doi: 10.13189/ujcmj.2019.070102.