Radon (for short; Ra) is a radioactive gas that "Produce natural from decaied of uranium (for short; U) (238U) to lead-206 (206Pb). U decomposed through a numbers of steaps to radium-226 (226Ra),Ra will then decomposes into anuther radioective elementes called radon gas(for short; Rg). As a Rg is colorless gas, odorles gas and tasteles gas. The prinsabel expoter of radun is U which is may come from the drink water, foods and same medicines from natural processed or as a result of human Activity [1]. New mathematical (for short;math) models have been constructed to estimate the construct a math models whose purpose to measure the Rg concentration in twenty types of medicine". The calculations are obtained using the numerical analysis methods.

The numerical analysis is considered as one of the important branches of math, which links between the analytical math and computer. It is used to find solution to some problems that cannot be solved by the analytical math.

In this study we used some types Numerical Analysis Method such as Neville's (for short; Ne), Spline (for short; Sp) method to conduct the calculation and construct the math models. These methods have helped us to obtain approximate calculations that are close to the previous studies, "There have been many study that have preceded this studies and it is consider as abstract study. Gassan Arif et al. study the mathl model to calcolate U Concentration in Urine sample of factory worker are due to the numbers of Years of work [2]. Ammar A. Battawy et al. A study the measuring the concentretion of Rg in diffarent Iraq radietion land [3]. In addition, Dr. Gassan Arif and Al-Douri, Ya did a study on math model of the physical properties of hexaigonal dimers [4]. Hameed et al. study to quantify the effect of potassiom radiation in as oil us the Ne and Hermate Numerical Methods [5]. In addition, in (2017) Dr. Gassan Arif et al. measuring the effect of U Radiution on worker in selected chemicals plants based on sp Numerical Analysis Method [6].

Table 1: Calculating M via the Ra and comparing them with the Obtained Exparimental Values, using Ne Method

Gassan Arif et al. Presenting a studies on the formation of math model to estimate the volumetric parameter based on the luttice constants through the use of Numerical Methods [7]".

Neville's Method [8,10]: "The essential idea of Ne Method is to approuximate the Value of a polynumial at a specific pointes without have to find all the coefficients of the Polynomial. The Ne Method can be defined as:

Let f be a function whose value ​​are at n points y₀, y₁,..., y_n are known. Let {n₁, n₂, . . . ,n} be the Set of distienct intagers k of set {0, 1, 2, . . . , n}. Let Pn₁, n₂, ..., n_k (y) standes for the lagrange poulynomial(for short; Lp) that agrees with the functions f at the k poiants yn₁, yn₂, . . ., yn_k, i.e., Pn₁, n₂, ..., n_k (yn₁) = f(yn₁), Pn₁, n₂, ..., nk (yn₂) = f(yn₂), . . . , Pn₁, n₂, ..., nk (yn_k) = f(ynk). Naturally, Pn₁, n₂, ..., n_k(y) Is the t only Polynomial of degree (k − 1) that pesses through the k points (yn₁, f(yn₁)), . . ., (yn_k, f(yn_k)). The ideas of the Ne method is to use Lp of lower Powers Recursively" In order to calculation Lp of high power relationship This is helpful, for Example, if you have the Lp base on some collection of Data point (yi , f(y_k)), k = 0, 1, . . . , m and we get a New data Point, (ym+1,f(ym+1)).

Establishing Math Models to Estimate the Calculations of M via Ra by Using Neville's Method

We will study the Ra in M the Mass of the solids sample in kg.''When us Ne method, the base can be written as follows:

Table 2: Calculating M via the Ra and Comparing is them with the Obteined Experimeuntal Value, using Sp Method

Figure 1: Comperison betwean the Obteined Result of Calculating M in Term of Ra and the Theuretical Result, us Ne Method

Figure 2: Comparison betwean the Obteined result of Calculating M in term of Ra and the Theuretical Resuelts, us sp Method

Figure 1 shows a comparison between the Ra, which is determined by Utilizing equation (1), and another exp. Ra by using Ne method.

Spline Method [9] 

"In math, a sp private Function Defined by Polynomials. In insertion problem, sp insertion is often preferred to Polynomial insertion because it yield similar result, even when us low degree Polynomials, while Avoiding Range's Phenomenon for higher degrees.

In the Computer science subfields of Computer- aided design and Computer graphics, the Term sp more duplicates refers to a piece Polynomial (Parametric) cuerve. Sp are poepular curve in these subfields because of the simplicity of its construction, their facilitates and accuracy of evaluation and their ability to Approximate complex shape through curve and interactive curve design".

Establishing Math Models to Estimate the Calculations of M via Ra by Using Spline Method

We will study The Radon in M the Mass of the solid Sample in kg. When using Sp meth, the rule can be is writ as follows:

Figure 2 shows a Comparison between the Ra, which is determined by utilizing, Equation (2) and another exp. Ra by using Spline method.

"When implementation the proposed math models, we obtained ideal and identical results, such as Applying two Numerical Analysis Methods (Ne and Sp methods). The ideal math models obtained from the two method is helped 

us Estimate the calculation the Concentration of Radon Gas in Different Medical Samples It has been shown that mathematical models were formulated using numerical analysis methods, Through the two methods, I concluded that namely Sp method in Table 2 is better than Ne method in Table 1 to get the estimated calculations, They were identical to the Experimental values ​​and the Error rates were very small and Almost non-existent".

1. Younis, M.A. et al. “Measurement of radon-222 exhalation rate in different kinds tablet medicine samples by detectors CR-39.” Journal of Kerbala University, vol. 10, no. 2, 2012.

2. Abumurad, K. et al. “Radiation protection dosimetry.” Radiation Protection Dosimetry, vol. 69, 1997, pp. 221–226.

3. Lohr, S.L. Sampling: Design and Analysis. Chapman and Hall/CRC, 2019.

4. Mansur, H.H. et al. “Radiation measurement.” Radiation Measurement, vol. 40, 2005, pp. 544–547.

5. Hameed, R.A. et al. “Estimating the amount of potassium radiation effect on soil using Neville’s and Hermite numerical methods.” Tikrit Journal of Pure Science, vol. 22, no. 9, 2017, pp. 100–105.

6. Arif, G.E. et al. “Estimating the amount of uranium radiation effect on the workers in selected chemical factories by using the numerical spline method.” Tikrit Journal of Pure Science, 2017.

7. Battawy, A.A. et al. “Tikrit Journal of Pure Science.” Tikrit Journal of Pure Science, vol. 21, 2018, pp. 147–153.

8. Lyness, J.N. and C.B. Moler. “Van der Monde systems and numerical differentiation.” Numerische Mathematik, vol. 8, 1966.

9. Ahlberg, J.H et al. The Theory of Splines and Their Applications. 1967.

10. Press, W. et al. “Polynomial interpolation and extrapolation.” Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Cambridge University Press, 1992, §3.1.