On Schlomilch's Integral Equation for the Ionospheric Plasma

De, S. S.; Sarkar, Biman; Mal, Manasi; De, Ma; Ghosh, Basudev; Adhikari, S. K.

doi:10.1143/jjap.33.4154

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…Unz (1966) and Gething and Maliphant (1967) used the Schlömilch's integral equations to described the ionosphere electron profile in a closed form. De et al (1994) derived an analytic form of the absorption index of the ionospheric plasma. They used it to build up the Schlomilch's integral equations.…”

Section: Introductionmentioning

confidence: 99%

Two efficient methods for solving Schlömilch’s integral equation

AL-Jawary¹,

Radhi

Ravnik

2017

IJICC

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Purpose-In this paper, the exact solutions of the Schlömilch's integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlömilch'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. Design/methodology/approach-First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch'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 regularizationvariational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method. Findings-The implementation of the two methods demonstrates the usefulness in finding exact solutions. Practical implications-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. Originality/value-In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlömilch'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 Schlömilch'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.

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Section: Introductionmentioning

confidence: 99%

Two efficient methods for solving Schlömilch’s integral equation

AL-Jawary¹,

Radhi

Ravnik

2017

IJICC

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“…It is an important and useful equation in atmospheric and terrestrial physics. For the derivation and applications of the equation in physics, we refer the reader to [1][2][3][4][5][6][7]. The standard linear Schlömilch's integral equation has an unknown function with a complicated argument and admits the following form: It's been shown that this equation provides a solution which has the following form:…”

Section: Introductionmentioning

confidence: 99%

On the Solutions of Schlömilch's Integral Equations

Altürk¹

2017

Celal Bayar Üniversitesi Fen Bilimleri Dergisi

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The linear Schlömilch'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 Schlömilch'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.

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Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems

Parand

Delkhosh

2016

Afr. Mat.

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On Schlomilch's Integral Equation for the Ionospheric Plasma

Cited by 5 publications

References 13 publications

Two efficient methods for solving Schlömilch’s integral equation

Two efficient methods for solving Schlömilch’s integral equation

On the Solutions of Schlömilch's Integral Equations

Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems

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