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In mathematics, the **Thomas–Fermi equation** for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi,^{ [1] }^{ [2] } which can be derived by applying the Thomas–Fermi model to atoms. The equation reads

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In mathematics, an **ordinary differential equation** (**ODE**) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term *ordinary* is used in contrast with the term partial differential equation which may be with respect to *more than* one independent variable.

**Llewellyn Hilleth Thomas** was a British physicist and applied mathematician. He is best known for his contributions to atomic physics and solid-state physics, in particular:

subject to the boundary conditions

If approaches zero as becomes large, this equation models the charge distribution of a neutral atom as a function of radius . Solutions where becomes zero at finite model positive ions.^{ [3] } For solutions where becomes large and positive as becomes large, it can be interpreted as a model of a compressed atom, where the charge is squeezed into a smaller space. In this case the atom ends at the value of for which .^{ [4] }^{ [5] }

Introducing the transformation converts the equation to

This equation is similar to Lane–Emden equation with polytropic index except the sign difference. The original equation is invariant under the transformation . Hence, the equation can be made equidimensional by introducing into the equation, leading to

In astrophysics, the **Lane–Emden equation** is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden. The equation reads

so that the substitution reduces the equation to

If then the above equation becomes

But this first order equation has no known explicit solution, hence, the approach turns to either numerical or approximate methods.

The equation has a particular solution , which satisfies the boundary condition that as , but not the boundary condition *y*(0)=1. This particular solution is

Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.^{ [6] } If the transformation is introduced, the equation becomes

**Arnold Johannes Wilhelm Sommerfeld**, was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored a large number of students for the new era of theoretical physics. He served as doctoral supervisor for many Nobel Prize winners in physics and chemistry.

The particular solution in the transformed variable is then . So one assumes a solution of the form and if this is substituted in the above equation and the coefficients of are equated, one obtains the value for , which is given by the roots of the equation . The two roots are . Since this solution already satisfies the second boundary condition, to satisfy the first boundary condition one writes

The first boundary condition will be satisfied if as . This condition is satisfied if and since , Sommerfeld found the approximation as . Therefore, the approximate solution is

This solution predicts the correct solution accurately for large , but still fails near the origin.

Enrico Fermi ^{ [7] } provided the solution for and later extended by Baker.^{ [8] } Hence for ,

**Enrico Fermi** was an Italian and naturalized-American physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. He made significant contributions to the development of statistical mechanics, quantum theory, and nuclear and particle physics.

where .^{ [9] }^{ [10] }

It has been reported by Esposito^{ [11] } that the Italian physicist Ettore Majorana found in 1928 a semi-analytical series solution to the Thomas–Fermi equation for the neutral atom, which however remained unpublished until 2001.

Using this approach it is possible to compute the constant *B* mentioned above to practically arbitrarily high accuracy; for example, its value to 100 digits is .

In probability theory and statistics, the **exponential distribution** is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

In mathematics, a **recurrence relation** is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.

In mathematics, for a given complex Hermitian matrix *M* and nonzero vector *x*, the **Rayleigh quotient** , is defined as:

In mathematics, the **classical orthogonal polynomials** are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials.

In mathematics, **separation of variables** is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

In mathematics and its applications, a classical **Sturm–Liouville theory**, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form

In mathematics, a **Cauchy-Euler equation** is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an *equidimensional* equation. Because of its particularly simple equidimensional structure the differential equation can be solved explicitly.

The **Pearson distribution** is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.

In numerical analysis, the **Crank–Nicolson method** is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time and can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century.

In general relativity, a **geodesic** generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

A differential equation can be **homogeneous** in either of two respects.

The **history of Lorentz transformations** comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product .

In mathematics, the **spectral theory of ordinary differential equations** is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In nonideal fluid dynamics, the **Hagen–Poiseuille equation**, also known as the **Hagen–Poiseuille law**, **Poiseuille law** or **Poiseuille equation**, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by George Stokes in 1845.

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. A **matrix differential equation** contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.

The **vibration of plates** is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

In fluid dynamics, a flow with periodic variations is known as **pulsatile flow**, or as **Womersley flow**. The flow profiles was first derived by John R. Womersley (1907–1958) in his work with blood flow in arteries. The cardiovascular system of chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in engines and hydraulic systems, as a result of rotating mechanisms pumping the fluid.

In applied mathematics, the **Atkinson–Mingarelli theorem**, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators.

The **critical load** is the maximum load which a column can bear while staying straight. It is given by the formula:

The **Fokas method**, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.

- ↑ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
- ↑ Bender, Carl M., and Steven A. Orszag. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.
- ↑ pp. 9-12, N. H. March (1983). "1. Origins – The Thomas–Fermi Theory". In S. Lundqvist and N. H. March. Theory of The Inhomogeneous Electron Gas. Plenum Press. ISBN 978-0-306-41207-3.
- ↑ March 1983, p. 10, Figure 1.
- ↑ p. 1562, R. P. Feynman, N. Metropolis, and E. Teller. "Equations of State of Elements Based on the Generalized Thomas-Fermi Theory".
*Physical Review***75**, #10 (May 15, 1949), pp. 1561-1573. - ↑ Sommerfeld, A. "Integrazione asintotica dell’equazione differenziale di Thomas–Fermi." Rend. R. Accademia dei Lincei 15 (1932): 293.
- ↑ Fermi, Enrico. "Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente." Zeitschrift für Physik A Hadrons and Nuclei 48.1 (1928): 73–79.
- ↑ Baker, Edward B. "The application of the Fermi–Thomas statistical model to the calculation of potential distribution in positive ions." Physical Review 36.4 (1930): 630.
- ↑ Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111], Francisco M.Fernández,
*Physics Letters A***372**, 28 July 2008, 5258-5260, doi : 10.1016/j.physleta.2008.05.071. - ↑ The analytical solution of the Thomas-Fermi equation for a neutral atom, G I Plindov and S K Pogrebnya,
*Journal of Physics B: Atomic and Molecular Physics***20**(1987), L547, doi : 10.1088/0022-3700/20/17/001. - ↑ Esposito, Salvatore (2002). "Majorana solution of the Thomas-Fermi equation".
*American Journal of Physics*.**70**: 852. arXiv: physics/0111167 . doi:10.1119/1.1484144.

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