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The integral of the standard normal distribution function is an integral without solution and represents
the probability that an aleatory variable normally distributed has values between zero and

Normal distribution is considered as one of the most important distribution functions in statistics because it is simple to handle analytically, that is, it is possible to solve a large number of problems explicitly; the normal distribution is the result of the central limit theorem. The central limit theorem states that in a series of repeated observations, the precision of the approximation improves as the number of observations increases [

The Homotopy Perturbation Method [

In this work the HPM method is applied to a problem having initial conditions. Nevertheless, it is also possible to apply it to the case where the problem has boundary conditions; here, the differential equation solutions are subject to satisfy a condition for different values of the independent variable (two for the case of a second order equation) [

This paper is organized as follows. In Section

This section deals with the solution of the Gaussian distribution integral; it is represented as follows:

This integral can be reformulated as a differential equation

Evaluating the integral (

To apply the homotopy perturbation method, the next equation is created

Also, the initial approximation for homotopy would be

Now, by the HPM it is assumed that (

Adjusting

Substituting (

Also, it is known that solution for (

Figure

(a) Gaussian distribution integral and approximation. (b) Relative error.

In Figure

(a) (

From this approximation it is possible to generate normal distribution functions like the error function or the cumulative distribution function.

From (

Now, function (

Next, three different approximations for the error function are presented; all of them are compared to the approximations proposed in this work ((

(1) In [

Due to the fact that (

(2) In [

The approximation for (

In order to compare our work with (

Replacing

for solving

From (

(3) In [

Figure

(a) Error function

The cumulative distribution function [

From (

Then, the process applied to (

Figure

(a) Cumulative distribution function (

In [

Gaussian function-related integrals.

Integral of Gaussian function | Approximate and exact solution | Relative error figure |
---|---|---|

(T.1) | ||

(T.2) | ||

(T.3) | ||

(T.4) | ||

(T.5) | ||

(T.6) | ||

(T.7) | ||

(T.8) | ||

(T.9) | ||

(T.10) | ||

(T.11) | ||

(T.12) | ||

(T.13) | ||

(T.14) | ||

(T.15) | ||

(T.16) |

For illustrative purposes, we have chosen values for

As an example to apply the error function, two cases are considered for the unidimensional heat flow equation.

Consider the case for a thin semi-infinite bar (

Scheme for Example 1.

From heat flow theory, it is known that

Replacing (

From boundary conditions follows that

now, it is possible to express the heat distribution as follows:

using

It is possible to express (

Expressing (

Figure

(a) Graphical comparison between exact (

Time range between 0 s and 2000 s

Time range between 0 s and 250 s

Another interesting example of heat flow is the nonstationary flux in an agriculture field due to the sun (Figure

Scheme for Example 2.

Let the origin be on the surface of the field, in such a way that the positive end for

This substitution immediately converts (

Integrating (

Using (

Determining

The approximate solution is obtained by substituting (

Figure

(a) Graphical comparison between exact (

Time range between 0 s and 1800 s

This paper presents the normal distribution integral as a differential equation. Then, instead of using a traditional linear function

The approximate solution of the cumulative distribution function was used to solve some defined and undefined integrals without known analytic solution, showing a low order relative error. Besides, the approximate error function (

Approximations (

The hyperbolic tangent was implemented in analog circuits [

Arctangent function was implemented in [

To multiply a current by a factor, it is only necessary to use current mirrors [

The addition or subtraction of constant values is achieved by connecting to the terminal a positive or negative current, respectively [

The addition of constants is equivalent to add constant current sources by using current mirrors [

Finally, after analyzing qualitatively the proposed approximation of this work, it is possible to establish a better approximation (see Figure

Relative error for approximation (

This work presented an approximate analytic solution for the Gaussian distribution integral (by using HPM method), the error function and the cumulative normal distribution providing low order relative errors. Besides, the relative error for the error function is comparable to other approximations found in the literature and has the advantage of being a simple expression. Also, it was possible to solve, satisfactorily, a series of normal distribution-related integrals, which may have potential applications in several areas of applied sciences. It was also demonstrated that the approximate error function can be employed on the practical solution of engineering problems like the ones related to heat flow. Besides, given the simplicity of the approximations, these are susceptible of being implemented in analog circuits focusing on the analog signal processing area.

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