Journal of Applied Mathematics

Volume 2015, Article ID 405108, 9 pages

http://dx.doi.org/10.1155/2015/405108

## Nonlinearities Distribution Homotopy Perturbation Method Applied to Solve Nonlinear Problems: Thomas-Fermi Equation as a Case Study

^{1}Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, VER, Mexico^{2}Equipe de Physique des Dispositifs à Semiconducteurs, Faculté des Sciences de Tunis, Tunis El Manar University, 2092 Tunis, Tunisia^{3}National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Santa María Tonantzintla, 72840 Puebla, PUE, Mexico^{4}Civil Engineering School, Universidad Veracruzana, Venustiano Carranza S/N, Colonia Revolucion, 93390 PozaRica, VER, Mexico^{5}Department of Electronics Engineering, Universidad Veracruzana, Venustiano Carranza S/N, Colonia Revolucion, 93390 Poza Rica, VER, Mexico

Received 3 November 2014; Revised 23 December 2014; Accepted 27 December 2014

Academic Editor: Shiping Lu

Copyright © 2015 U. Filobello-Nino 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

We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results.

#### 1. Introduction

The numerical simulation for the charge distribution and the electric field inside an atom is a very difficult task, especially for complex atoms. For these cases, the Thomas-Fermi (T-F) equation can be employed in order to obtain highly accurate approximate results. T-F method is based on the fact that most of complex atoms, with a large number of electrons, have relatively large quantum numbers, and therefore the semiclassical approach can be employed. Then, it is possible to apply the concept of cells in the phase space to the states of individual electrons [1].

The Thomas-Fermi equation describes mathematically an infinite atom, without border; however the model is not applicable to both distances from the nucleus: too small and too large. More precisely, its application domain is limited to a range of distances : large relative to the quantity and small compared to 1, where is the atomic number of the atom. However, for complex atoms, most of the electrons are in that interval, for which the study of T-F equation becomes important.

One of the important consequences of the Thomas-Fermi theory follows from the above. The atom has an outer boundary for values of ; thus the theory predicts that the dimensions of the atom are independent of , and for the same reason the ionization energy of the outer electrons is also independent of the atomic number [1].

Problems like the one mentioned give rise to the search of solutions to nonlinear differential equations but, unfortunately, solving this kind of equations is a difficult task. As a matter of fact, most of the time, only an approximate solution to such problems can be got. In order to approach various types of nonlinear differential equations, several methods have been proposed such as those based on variational approaches [2–4], tanh method [5], exp-function [6], Adomian’s decomposition method [7–12], parameter expansion [13], homotopy perturbation method [14–27], homotopy analysis method [25, 28–32], and perturbation method [28, 33, 34], among others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally [35].

This study proposes a variation of the homotopy perturbation method (HPM), by using nonlinearities distributions [17], which allows finding an approximate solution of Thomas-Fermi equation [23, 36–41]. On the other hand, it will be seen that it is convenient to introduce an adjusting parameter in order to enlarge the domain of convergence of our approach.

This paper is organized as follows. In Section 2, we provide the basic concept of nonlinearities distributions homotopy perturbation method (NDHPM). Section 3 presents the application of NDHPM to find an approximate solution of Thomas-Fermi equation. Section 4 discusses the main results obtained. Finally, a brief conclusion is given in Section 5.

#### 2. Basic Idea of NDHPM

The standard homotopy perturbation method (HPM) was proposed by Ji Huan He. It was introduced like a powerful tool to approach various kinds of nonlinear problems. The homotopy perturbation method can be considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), and it is not restricted to small parameters as traditional perturbation methods. For example, HPM method requires neither small parameter nor linearization but only few iterations to obtain accurate solutions.

To figure out how HPM method works, consider a general nonlinear equation in the form with the following boundary conditions: where is a general differential operator, is a boundary operator, a known analytical function, and is the domain boundary for . can be divided into two operators and , where is linear and nonlinear; from this last statement, (1) can be rewritten as Generally, a homotopy can be constructed in the form Or where is a homotopy parameter, whose values are within range of 0 and 1; is the first approximation for the solution of (3) that satisfies the boundary conditions.

Assuming that solution for (4) or (5) can be written as a power series of , Substituting (6) into (5) and equating identical powers of terms, there can be found values for the sequence .

When , it yields the approximate solution for (1) in the form Another way to build a homotopy, which is relevant for this paper, it is by considering the following general equation: where and are the linear and nonlinear operators, respectively. It is desired that solution for describes, accurately, the original nonlinear system.

By the homotopy technique, a formulation is constructed as follows [16]: Again, it is assumed that solution for (9) can be written in the form (6); thus, taking the limit when results in the approximate solution of (8). The convergence of solutions obtained by HPM method is discussed in [20, 21, 25, 26].

A recent report [17] introduced a modified version of homotopy perturbation method, which eases the solutions searching process for (3). As first step, a homotopy of the following form is introduced: It can be noticed that the homotopy function (10) is essentially the same as (4), except for the nonlinear operator and the nonhomogeneous function , which contain embedded the homotopy parameter . We propose first to subdivide and into a series of terms and multiply by the most nonlinear terms, where is an integer number greater or equal to zero. The power is selected according to how much displacement is desired in the interactions for the corresponding nonlinear term of or . The introduction of that parameter within the differential equation is a strategy to redistribute the nonlinearities between the successive iterations of the HPM method, and thus it increases the probabilities of finding the sought solution. The rest of the method is exactly the same as the standard procedure for the HPM. Therefore, we establish that When , it results in the approximate solution for (3) in the form An advantage of this procedure is that, given the distribution of nonlinearities, from the differential equation, over the successive iterations of (11), less complex analytic approximations may be obtained than those generated by the original standard HPM.

Finally, convergence of solutions obtained by NDHPM method is discussed in [17].

#### 3. Approximate Solution of Thomas-Fermi Equation

In order to facilitate understanding of the NDHPM method, we will solve approximately the equation where prime denotes differentiation with respect to .

is a function related to the electrostatic potential inside the atom, is a variable proportional to distance from the nucleus , and the most accurate value of is [36].

The above equation is a modified version of the original problem of Thomas-Fermi, where (13) is subject to the boundary conditions , [28, 41] although there are other conditions that can be applied [40].

Following references [16, 24] instead of defining a linear and nonlinear part in the above equation, we add and subtract as it is shown, in order to add a term to linear operator to facilitate the search for convergent solutions; thus (13) can be rewritten as where and are constant parameters to determine.

The linear part is identified as and the nonlinear is In order to simplify the calculations, we will redistribute the nonlinear term by using NDHPM method, starting from (9), in the form by substituting into (17), and equating terms having identical powers of we obtain

In this paper we study the first order approximation, (19)-(20). We will see that the nonlinear term of (see (16)) contains the sufficient information to obtain a good analytical approximation for (13).

The solution of (19) that satisfies the initial conditions is given by where and are constants related to and .

By substituting (21) into (20) we obtain In order to simplify (22) and obtain a handy approximation for (13), we will assume that there is an adequate choice of and such that , valid for ; in such a way we can rewrite (22) approximately as where we have defined This assumption is justified later.

To solve (23), we employ the method of variation of parameters [42] which requires evaluating the following integrals: where and are the solutions of the homogeneous differential equation is the Wronskian of these two functions, which is given by and is the right-hand side of (23).

Substituting and (27) into (25) leads to By substituting , the above equations take the form Therefore, the solution of (23) is written according to the method of variation of parameters as [42] where and are constants and and are given by (29) and (30), respectively.

Applying the initial condition to (31) leads to Equation (32) is simplified to On the other hand, to apply the condition , we differentiate (31) to obtain After performing algebraic simplifications to (34), we obtain Applying the condition to (35) the following is obtained: since and (see (29) and (30)).

From (33) and (36) we obtain and since ; therefore (31) becomes By substituting (21) and (37) into (12), we obtain the following first order approximation for the solution of (13): We will show later that it is possible to achieve a good fit and enlarge the domain of convergence for (38) (keeping at the same time the handy character of the proposed solution), by introducing in (38) an adjusting parameter (consistent with the initial conditions), as it is shown in The inclusion of parameter, , is motivated because the solution for the homogeneous equation (19) is undetermined by a constant factor (without considering the initial values) and from the following argument.

Let , since , ; the first order solution for a function, say , which defines a slightly different problem from the original, is given by or , where , , , and . Thus, after dividing by , is obtained (where , and we have supposed that for some problems . As a matter of fact, following (21)–(30) we conclude that for this case ; therefore .

Substituting (29) and (30) into (39) we obtain In [14] a high accurate approximation of normal distribution integral was reported, which let us write the above integrals as Note that is not a Gaussian integral on the assumption that for , since it implies that .

Equations (41) allow us to write (40) as To transform (43) into an analytical expression, we evaluate (42) keeping terms up to the ninth power, to obtain the following compact fractional power function: where the main contribution of this approximation to (43) is in the range of because it is multiplied by a negative exponential (see (43) and Section 4).

Therefore, substituting (44) into (43), the following is obtained: In order to obtain a good approximation from (45), we optimize the values of the aforementioned parameters as follows: , , and using the procedure reported in [14, 17, 27] (note that with these values, the assumption is satisfied, besides the above results allow to know and whose values are not needed in the search for the solution of the system (19) and (20)).

Figure 1 and Table 1 show the comparison between approximate solution (45) and the exact solution. The accuracy of (45) is clear as an approximate solution for (13).