Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 3081264, 13 pages

https://doi.org/10.1155/2019/3081264

## A Mathematical Approach with Fractional Calculus for the Modelling of Children’s Physical Development

^{1}Informatics Institute, Istanbul Technical University, Istanbul 34467, Turkey^{2}Medicine Faculty, Istanbul Medeniyet University, Istanbul 34000, Turkey

Correspondence should be addressed to Nisa Özge Önal; rt.ude.uti@61lano

Received 6 February 2019; Accepted 11 August 2019; Published 12 September 2019

Academic Editor: Didier Delignières

Copyright © 2019 Nisa Özge Önal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

From birth to now, it is getting more and more important to keep track of the children development, because knowing and determining the factors related to the physical development of the children would provide better and reliable results for children care. In this study, we developed a mathematical approach to have the ability of analysing and examining factors such as weight, height, and body mass index with respect to the age. We used 7 groups for weight, height, and body mass index in Percentage Chart of Turkey. We developed a continuous curve which is valid for any time interval by using discrete weight, height, and body mass index data of 0–18 years old children and the least squares method. By doing so, it became possible to find the percentage and location of the children in Percentage Chart. We advanced a new mathematical model with the help of fractional calculus theory. The results are quite successful and better compared to linear and Polynomial Model analysis. The method provides the opportunity to predict expected values of the children for the future by using previous data obtained in the development of the children.

#### 1. Introduction

Fractional integral and derivative, are distinctly defined as the derivative and integral with non-integer order. The concept of fractional calculus, which means in more general form, the calculus of integrals and derivatives of any arbitrary real or complex order, is built up from a question raised in 1695 by French Mathematician Marquis de L’Hôpital (1661–1704) to Wilhelm Leibniz (1646–1716) [1–3]. He asked what if derivative order becomes 0.5, and Leibniz’s response to this question was “this is an apparent paradox from which, one day, useful consequences will be drawn …” [1–6].

For 50 years, many mathematicians, engineers, scientists, and researchers prove in their studies that, fractional derivatives and integrals contain important information about the systems that they are searching for. Especially, the fractional derivative provides pretty good insight for the memory and hereditary of a process or phenomena. The fractional calculation is widely used in the control theory, mechanics, and economics, finance, electromagnetic and mostly in biology [7–18]. In this study, we advanced a mathematical approach which analyses and examines the factors continuously related to the physical development of the children by the help of fractional calculus theory. We used 7 groups for body weight, height, and body mass index in Percentage Chart of Turkey. We developed a continuous curve valid for any time interval by using discrete weight, height, and body index data of 0–18 years old children and least squares method. Therefore, it became possible to find the percentage and location of the children in Percentage Chart. The results of the fractional calculus model analysis are more successful than the linear and Polynomial Model analysis. The method provides the opportunity to predict expected values of the children for the future by using previous data obtained in the development of the children. In this paper, the theory, numerical results of the developed theory and comparison with the other modelling methods such as linear and polynomial methods are presented.

#### 2. Formulation of the Problem

Firstly, fractional derivative determined from the Riemann-Liouville equation [5] which has the form;

Here, is a Gamma Function which is defined as . The Fractional Order (FO), varies from 0 to 1.

In order to improve the convergence of the Polynomial Model results, we utilized the theory of fractional calculus [4, 5]. The vital question we asked in this paper is what if the fractional derivative of is equal to the expression given in equation (2). Derivative order is and (0, 1).

Here, corresponds to the data of children’s weight, height, and body mass index with respect time which is denoted as *x* in equation (2).

After, Laplace transform of equation (2) is taken [4]. stands for the Laplace Transform and stands for the inverse Laplace transform. Laplace transform of is denoted as . Inverse Laplace transform of equation (3) is given as

As mentioned in the introduction part, our purpose is to model the children’s physical development with respect to time by using previously found data, and we use the least square mean method to achieve our goal [19, 20]. Due to having the finite number of discrete data, summation corresponds to in equation (3) also needs to be truncated to . Truncated version of equation (4) is given in

We have a dataset to make regression on it.

The dimension of the dataset is . In the dataset, the corresponding value for each is given as . Here, represents the time, and corresponds to the weight, height, and body mass index of the child in the specific time. The dimension of the dataset determines the upper limit of value given in equation (5) by the nature of solving the System of Linear Algebraic equations (SLAE) [19].

The error between the value and is showed as in equation (7). In the least squares method, the purpose is to minimize the square of the total error contributing from each data points.

In equation (8), summation of error’s square is given.

In order to minimize the total error given in equations (8) and (9) needs to be satisfied [19].

After finding equations (8) and (9), following SLAE is achieved. SLAE can be denoted aswhere,

Here, is the matrix transpose. Unknown coefficients in the vector can be found by equationwhere, is the inverse of matrix.

#### 3. Dataset

In this study, we use body weight, height, and body mass index of 0–18 years old children indicated in Percentage Chart of Turkey. The dataset includes 7 groups (3-10-25-50-75-85-97 percentiles for body weight and height, 5-15-25-50-75-85-95 percentiles for body mass index) for boys and girls. See Tables S1–S3 in the Supplementary Material [20, 21].

#### 4. Results and Discussion

In this study, Polynomial Model and Fractional Model were used for comparing different exponent values up to 3, 4, and 5 in equation (5);

We obtained results from Linear, Polynomial, and Fractional Model respectively. Mean Absolute Percentage Error (MAPE) was used for comparing the models [19]. MAPE formulation is showed in equation (13). Tables 1–3 illustrates the results of the age versus body weight, age versus height, and age versus body mass index, respectively.where is real value and is predicted value.