scholarly journals Approximate Solution of the Thomas–Fermi Equation for Free Positive Ions

Atoms ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 87
Author(s):  
Aleksey A. Mavrin ◽  
Alexander V. Demura
Keyword(s):  
Approximate Solution ◽  
Asymptotic Representation ◽  
High Accuracy ◽  
Universal Functions ◽  
Ionization Degree ◽  
Thomas Fermi ◽  
Positive Ions ◽  
Ion Size

The approximate solution of the nonlinear Thomas–Fermi (TF) equation for ions is found by the Fermi method. The solution is based on the new asymptotic representation of the TF ion size valid for any ionization degree. The two universal functions and their derivatives, introduced by Fermi, are calculated by recent effective algorithms for the Emden–Fowler type equations with the accuracy sufficient for majority of applications. The comparison of our results with those obtained previously shows high accuracy and validity for arbitrary values of ionization degree. This study could potentially be of interest for the statistical TF method applications in physics and chemistry.

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

An optimal iteration method with application to the Thomas-Fermi equation

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Approximate Solution ◽  
Approximate Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Analytical Approximate Solution ◽  
Thomas Fermi ◽  
Strongly Nonlinear ◽  
Good Agreement

AbstractThe aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.

scholarly journals Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
M. Tavassoli Kajani ◽  
S. Vahdati ◽  
Zulkifly Abbas ◽  
Mohammad Maleki
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Galerkin Method ◽  
High Accuracy ◽  
Integrodifferential Equations ◽  
Expansion Test ◽  
Infinite Intervals ◽  
System Of Integrodifferential Equations ◽  
Rational Chebyshev ◽  
Chebyshev Functions

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

scholarly journals On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 685-689 ◽  
Author(s):  
Ali Akgül ◽  
Dumitru Baleanu ◽  
Mustafa Inc ◽  
Fairouz Tchier
Keyword(s):  
Approximate Solution ◽  
Fractional Differential Equations ◽  
Numerical Experiments ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
High Accuracy ◽  
Reproducing Kernel Method ◽  
Fractional Differential ◽  
Kernel Model ◽  
Theoretical Results

AbstractIn this manuscript we investigate electrodynamic flow. For several values of the intimate parameters we proved that the approximate solution depends on a reproducing kernel model. Obtained results prove that the reproducing kernel method (RKM) is very effective. We obtain good results without any transformation or discretization. Numerical experiments on test examples show that our proposed schemes are of high accuracy and strongly support the theoretical results.

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