On Schlomilch's Integral Equation for the Ionospheric Plasma
TL;DR: The analytical form of the absorption index appropriate to the D- and E-region of the ionospheric plasma is derived in this article. And the results are presented graphically along with an earlier work.
Abstract: The analytical form of the absorption index appropriate to the D- and E-region of the ionospheric plasma is derived. It has been utilised to build up Schlomilch's integral equation. The corresponding solution is obtained in terms of certain ionospheric parameters. The variation of effective collision frequency with height has been computed numerically from the derived solution. The results are presented graphically along with an earlier work.
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TL;DR: In this paper, the convergence of a Dickson polynomial solution of the model problem is investigated by means of the residual function, and the exact solutions are compared with other well-known methods in tables.
29 citations
01 Jun 2017
TL;DR: The generalized fractional order of the Chebyshev orthogonal functions (GFCF) collocation method is used to handle many forms of Schlomilch integral equations.
Abstract: In this paper, the linear and nonlinear Schlomilch’s integral equations and their generalized forms are studied. The Schlomilch’s integral equations are used for many ionospheric problems, atmospheric and terrestrial physics. The generalized fractional order of the Chebyshev orthogonal functions (GFCF) collocation method is used to handle many forms of Schlomilch’s integral equations. The GFCF method can be used in the applied physics, applied mathematics, and engineering applications. The reliability of the GFCF method is justified through illustrative examples.
20 citations
TL;DR: The accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.
Abstract: In this paper, the exact solutions of the Schlomilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlomilch’s integral equations have many applications in atmospheric, terrestrial physics and ionospheric problems. They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations. The paper aims to discuss these issues.,First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlomilch’s integral equation. Second, the authors implement the regularization method with the variational iteration method for the same purpose. The effectiveness of the regularization-Homotopy method and the regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.,The implementation of the two methods demonstrates the usefulness in finding exact solutions.,The authors have applied the developed methodology to the solution of the Rayleigh equation, which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering. These include the analysis of batch distillation in chemistry, scattering of electromagnetic waves in physics, isotopic data in contaminant hydrogeology and others.,In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlomilch’s integral models. Each method has been accompanied with the use of the regularization method. This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlomilch’s integral equation which is easy to implement. In addition to that, the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.
4 citations
Cites background from "On Schlomilch's Integral Equation f..."
...De et al. (1994) derived an analytic International Journal of Intelligent Computing and Cybernetics Vol. 10 No. 3, 2017 pp. 287-309 © Emerald Publishing Limited 1756-378X DOI 10.1108/IJICC-11-2016-0042 The authors would like to thank the anonymous referees, Managing Editor and Editor in Chief for…...
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TL;DR: In this article, an alternative closed-form expression for solutions of the linear and the nonlinear Schlomilch's integral equation in terms of the well-known gamma function is provided.
Abstract: The linear Schlomilch's integral equation is an important and useful equation in atmospheric and terrestrial physics. The equation and its solution have been used for some ionospheric problems. It can also be considered as a special type of Fredholm integral equation of the first kind. This correspondence allows one to use the mathematical tools available for solving Fredholm integral equation of the first kind. In this article, we provide an alternative closed-form expression for solutions of the linear and the nonlinear Schlomilch's integral equation in terms of the well-known gamma function. Some elaborate examples are provided to demonstrate the simplicity and applicability of the proposed formulae.
Cites background from "On Schlomilch's Integral Equation f..."
...For the derivation and applications of the equation in physics, we refer the reader to [1-7]....
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