A numerical method for solving some model problems arising in science and convergence analysis based on residual function

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.

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

The radiation from a point charge moving uniformly in a plasma is investigated when the charge is moving in the direction of an external magnetic field. In general, there are two modes, for each of which all the components of the electric and magnetic field are present. The two parameters of interest in this problem are the ratio of the velocity of the charges to the free-space velocity of electromagnetic waves, and the ratio of the gyromagnetic frequency to the plasma frequency of the electrons. For two sets of values of these parameters, the frequency and the angular spectrum of the emitted radiation are obtained. In certain cases, as many as three Cherenkov rays are found to propagate in the same direction; these multiple-rays, however, correspond to different frequency components and to different modes of propagation. The motivation for this investigation is indicated briefly. (auth)

RADIATION FROM A UNIFORMLY MOVING CHARGE IN AN ANISOTROPIC PLASMA. Technical Report No. 389

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.

Two efficient methods for solving Schlömilch’s integral equation

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.

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On the Solutions of Schlömilch's Integral Equations

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.

On Schlomilch's Integral Equation for the Ionospheric Plasma

A brief review of Cerenkov radiation within the upper atmospheric plasma has been presented. Different attempts in this context are systematically discussed. The results of analysis about the nature and characteristics of VLF hisses in terms of incoherent Cerenkov radiation are given in a concise manner. The occurrence of resonance cone has also been reported.

M. De

Papers

On Schlomilch's Integral Equation for the Ionospheric Plasma

Čerenkov radiation within the Earth's upper atmosphere