On HPM Approximations for the Cumulative Normal Distribution Function

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.

Perihelion Advance and Trajectory of Charged Test Particles in Reissner-Nordstrom Field via the Higher-Order Geodesic Deviations

“…In the previous section, we have calculated the trajectory of charged particles up to second-order. To find a more accurate trajectory, we need to obtain the higher-order terms of expansion (9). Using the first and second-order solutions and third-order equation (51)…”

“…In fact, we generalize the novel method used in reference [1] for neutral particles in the Schwarzschild metric to the charged particles in the Reissner-Nordstrom metric. Recently, an analytical computation of the perihelion advance in general relativity via the Homotopy perturbation method has been proposed in [9]. Also, one can study the perihelion advance of planets in general relativity and modified theories of gravity by using different methods in [10]- [24].…”

Studying gravitational deflection of light by Kiselev black hole via homotopy perturbation method

“…Recently there were studies in which this method was used for analytical computations in the field of cosmology and astrophysics (see, e.g. [36] - [39]. It should be also mentioned that applications of HPM can be found in the field of astrophysics in different contexts that creates a new research field [40,41].…”

Higher-order geodesic deviations and orbital precession in a Kerr–Newman space–time

“…The motion of neutral and charged test particles along closed orbits in Kerr-Newman metric has been analysed in an exhaustive manner in many papers [14]- [20]. For further insight on the alternative methods in calculating the orbital precession and planetary orbits in Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman space-times see [21]- [29].…”