Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

Lazhar Bougoffa; Smail Bougouffa; Ammar Khanfer; Lazhar Bougoffa; Smail Bougouffa; Ammar Khanfer

doi:10.3934/math.2023534

AIMS Mathematics

2023, Volume 8, Issue 5: 10529-10546. doi: 10.3934/math.2023534

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Research article

Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions

1.
Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
2.
Department of Physics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia

Received: 08 December 2022 Revised: 14 February 2023 Accepted: 15 February 2023 Published: 02 March 2023
MSC : 34B16, 34B40, 65N35

The existence and uniqueness theorem for the generalized boundary value problem of the Thomas-Fermi equation:

$ \begin{eqnarray*} \left\{ \begin{array}{l} y''+f(x, y) = 0, \ 0<x <\infty, \\ y(0) = 1, \ y(\infty) = 0, \end{array} \right. \end{eqnarray*} $

where

$ \begin{equation*} \label{6}f(x, y) = -y \left(\frac{y}{x}\right)^{\frac{p}{p+1}}, \ p>0, \ 0<x <\infty, \end{equation*} $

is proved. Also, highly accurate approximate solutions are obtained explicitly for this new boundary value problem which arises in particular studies of many-electron systems (atoms, ions, molecules, metals, crystals). To the best of our knowledge, the results obtained here are new and provide the lower and upper bounds approximate solutions for the generalized Thomas-Fermi problem.
- Thomas-Fermi equation,
- existence and uniqueness theorem,
- approximate solution,
- lower and upper bounds,
- Adomian decomposition method,
- Adomian polynomials
Citation: Lazhar Bougoffa, Smail Bougouffa, Ammar Khanfer. Generalized Thomas-Fermi equation: existence, uniqueness, and analytic approximation solutions[J]. AIMS Mathematics, 2023, 8(5): 10529-10546. doi: 10.3934/math.2023534

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Abstract

The existence and uniqueness theorem for the generalized boundary value problem of the Thomas-Fermi equation:

$ \begin{eqnarray*} \left\{ \begin{array}{l} y''+f(x, y) = 0, \ 0<x <\infty, \\ y(0) = 1, \ y(\infty) = 0, \end{array} \right. \end{eqnarray*} $

where

$ \begin{equation*} \label{6}f(x, y) = -y \left(\frac{y}{x}\right)^{\frac{p}{p+1}}, \ p>0, \ 0<x <\infty, \end{equation*} $

is proved. Also, highly accurate approximate solutions are obtained explicitly for this new boundary value problem which arises in particular studies of many-electron systems (atoms, ions, molecules, metals, crystals). To the best of our knowledge, the results obtained here are new and provide the lower and upper bounds approximate solutions for the generalized Thomas-Fermi problem.

References

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