Approximate rational solutions to the Thomas–Fermi equation based on dynamic consistency

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Abstract

We construct two rational approximate solutions to the Thomas–Fermi (TF) nonlinear differential equation. These expressions follow from an application of the principle of dynamic consistency. In addition to examining differences in the predicted numerical values of the two approximate solutions, we compare these values with an accurate numerical solution obtained using a fourth-order Runge–Kutta method. We also present several new integral relations satisfied by the bounded solutions of the TF equation.

Introduction

The major purpose of this paper is to construct a new rational approximation to the Thomas–Fermi equation d2yxdx2=yx32x12,with boundary conditions y(0)=1,y()=0.

This equation arises in the modeling of certain phenomena in the atomic structure of matter [1], [2].

Since it is expected that no exact analytical solution formula exists for the general solution in terms of a finite combination of elementary functions, a broad range of approximate solutions have been derived [3], [4], [5], [6], [7], [8], [9]. The mathematical procedures used to obtain these expressions include variational methods [3], [6], the use of iteration techniques [4], application of the Adomian decomposition method [5], homotopy analysis [7], and rational approximations [8]. The indicated references [3], [4], [5], [6], [7], [8], [9] are but a small subset of the hundreds of publications on this subject. Note that the existence of such a substantial research literature is due, in large part, to the fact that this thorny boundary-value problem has drawn out a significant number of ideas as to how approximations to its solutions should be done.

The purpose of this paper is to use a new methodology to construct approximate solutions to the Thomas–Fermi (TF) equation. We do this within the framework of the concept or principle of dynamic consistency [10] and the restrictions that it imposes on any mathematical representation of the forms used for rational approximations to a TF equation solution. We demonstrate that this is rather easy to do.

In outline, Section 2 provides a summary of several of the general, exact, and for the most part qualitative properties of the TF equation [11]. This section includes several new results of the authors. Section 3 introduces and defines the concept of dynamic consistency [10] and it is used in Section 4 to construct two elementary rational approximations to the TF solution. Finally, in Section 5, we provide a summary of our results and discuss possible extensions to this work.

Section snippets

Preliminaries: Exact results

It turns out that even in the absence of knowledge of an explicit, exact solution for the TF equation, many of the essential properties of such a solution can be derived. Below, we give a concise summary of these items and refer the reader to the paper of Hille [11] for proofs of some of these statements.

(a) An exact solution to the TF equation is yxysx=144x3.Note that it contains no arbitrary parameters and thus is not a special case of the (unknown) general solution. We denote it as, ysx and

Dynamic consistency

Consider two systems A and B. Let system A have the property P. If system B also has property P, then B is said to be dynamic consistent to A, with respect to property P [10].

Dynamic consistency  (DC) has served as one of the fundamental principles of the nonstandard, finite difference methodology for constructing improved discrete models for the numerical integration of differential equations [10]. It serves as an assessment of the “closeness” or “fidelity” of two systems based on their

Approximate solutions

We take the following two ansatzes to represent approximations to the solution of the TF equation ya1(x)=11+Bx+1144x3, ya2x=11+Bx43x32+Cx2+1144x3,C=B22, where B=1.588071, and as a consequence of (33), C=1.260985.

Close inspection of ya1(x) and ya2x shows that they are DC with the five conditions listed in Section 3. Observe that ya1x is a rational function of x, while ya2x is a rational function of x.

Fig. 2 gives plots of ya1(x) and ya2(x) vs. x. The two curves lie close to each other, but

Discussion

In summary, we have constructed two representations of approximate solutions to the TF equation. However, it must be stated that this equation is itself an approximation to an (unknown) equation describing atomic quantum phenomena [1], [2]. In addition, all numerical procedures have errors associated with their algorithms and the hardware on which they are implemented.

The methodology used to generate the ansatzes, given in Eqs. (31), (32), (33), is based on constructing for yx, the exact

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