scholarly journals About Solution of the Nonlinear Generalized Abel Integral Equation

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Power Function ◽  
Nuclear Physics ◽  
Abel Equation ◽  
Abel Integral Equation ◽  
Spatial Graphs

As is known, many problems of electronics, nuclear physics, optics and astrophysics, etc. are described by the Abel integral equation of the first kind. In this paper we consider the nonlinear generalized Abel equation and show that its solution can be represented as an integral of a power function. It is shown that the constructed analytical solution and the symbolic solution obtained by means of the computer mathematics system Maple coincides, and their planar and spatial graphs are presented.

scholarly journals Analytical solution of Abel integral equation arising in astrophysics via Laplace transform

2015 ◽  
Vol 23 (1) ◽  
pp. 102-107 ◽  
Author(s):  
Sunil Kumar ◽  
Amit Kumar ◽  
Devendra Kumar ◽  
Jagdev Singh ◽  
Arvind Singh
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Abel Integral Equation

scholarly journals On the Solution of Distributional Abel Integral Equation by Distributional Sumudu Transform

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Deshna Loonker ◽  
P. K. Banerji
Keyword(s):  
Integral Equation ◽  
Abel Equation ◽  
Abel Integral Equation ◽  
Sumudu Transform

Solution of the Abel integral equation is obtained using the Sumudu transform and further, distributional Sumudu transform, and, distributional Abel equation are established.

scholarly journals On the numerical solution of an Abel integral equation

1979 ◽  
Vol 16 (3) ◽  
pp. 497-503
Author(s):  
R. Smarzewski ◽  
H. Malinowski
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Abel Integral Equation

scholarly journals Newtonsche Aufheizung, Abelsche Integralgleichungen zweiter Art und Mittag-Leffler-Funktionen

1987 ◽  
Vol 42 (10) ◽  
pp. 1141-1146 ◽  
Author(s):  
Rudolf Gorenflo
Keyword(s):  
Integral Equation ◽  
Explicit Solution ◽  
Temporal Evolution ◽  
Plane Boundary ◽  
Qualitative Properties ◽  
Boundary Temperature ◽  
Abel Integral Equation ◽  
Homogeneous Half Space ◽  
Interior Boundary ◽  
The Difference

The problem of heating a homogeneous half-space by radiation from outside across the plane boundary is considered. Newtonian heating means that the heat flux across the boundary is proportional to the difference of outside temperature and interior boundary temperature. The outside temperature is assumed to be constant and positive, the initial inside temperature is zero everywhere. The problem is onedim ensional in space. The temporal evolution of the inward boundary temperature obeys an Abel integral equation of second kind for whose explicit solution three methods are described (one by Laplace transform , the other two by infinite series defining the Mittag-Leffler function of index 1/2). The explicit solution facilitates discussion of its qualitative properties. Finally, the general Abel integral equation of second kind is treated by Mittag-Leffler functions.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

Three and Four Centre Oscillators and Their Application in Nuclear Physics

1974 ◽  
Vol 29 (7) ◽  
pp. 1003-1010 ◽  
Author(s):  
Peter Bergmann ◽  
Hans-Joachim Scheefer
Keyword(s):  
Analytical Solution ◽  
Schrödinger Equation ◽  
Nuclear Physics ◽  
Schrodinger Equation ◽  
Numerical Procedure ◽  
Orthogonal Basis ◽  
Two Dimensional ◽  
Schmidt's Method ◽  
Symmetry Properties

The extension of the nuclear two-centre-oscillator to three and four centres is investigated. Some special symmetry-properties are required. In two cases an analytical solution of the Schrödinger equation is possible. A numerical procedure is developed which enables the diagonalization of the Hamiltonian in a non-orthogonal basis without applying Schmidt's method of orthonormalization. This is important for calculations of arbitrary two-dimensional arrangements of the centres.

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