© 2019 JETIR March 2019, Volume 6, Issue 3 www.jetir.org (ISSN-2349-5162)
JETIR1903492 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 600
A New Application of Mohand Transform for
Handling Abel’s Integral Equation
Sudhanshu Aggarwal
1*
, Swarg Deep Sharma
2
, Anjana Rani Gupta
3
1*
Assistant Professor, Department of Mathematics, National P.G. College, Barhalganj,
Gorakhpur-273402, U.P., India
2
Assistant Professor, Department of Mathematics, Nand Lal Singh College Jaitpur Daudpur Constituent of Jai Prakash University
Chhapra-841205, Bihar, India
3
Professor, Department of Mathematics, Noida Institute of Engineering & Technology, Greater Noida-201306, U.P., India
ABSTRACT: Abel’s integral equation is an important singular integral equation and generally appears in many branches of
sciences such as atomic scattering, mechanics, radio astronomy, physics, electron emission, X-ray radiography and seismology. In
this paper, we give a new application of Mohand transform for handling Abel’s integral equation and the complete procedure of
handling Abel’s integral equation using Mohand transform explain by giving some numerical applications in application section.
KEYWORDS: Abel’s integral equation, Mohand transform, Inverse Mohand transform, Convolution theorem.
AMS SUBJECT CLASSIFICATION 2010: 44A05, 34A12, 44A35.
I. INTRODUCTION: In 1823, Niels Henrik Abel discussed the motion of particle on smooth curve lying on a vertical plane
using Abel’s integral equation in mathematical form as [1-2]
() = ∫
1
√−
()
0
(1)
Here the kernel of integral equation, (, ) =
1
√−
becomes ∞ at = , the function () is known function and the function
() is unknown function.
In the modern time, integral transforms are widely used mathematical techniques for solving advanced problems of science and
engineering which mathematically express in terms of differential equations with constant or variable coefficients, partial
differential equations with constant or variable coefficients, integral equations, partial integro-differential equations, integro-
differential equations etc. Many researchers [3-30] applied different integral transforms (Laplace transform, Fourier transform,
Hankel transform, Kamal transfom, Elzaki transform, Mohand transform, Aboodh transform, Sumudu transform, Wavelet
transform etc) for solving many problems of science, engineering and daily life.
Mohand and Mahgoub [31] defined “Mohand transform’’ of the function () for ≥0 in the year 2017 as
{()} =
2
∫ ()
−
∞
0
= (),
1
≤≤
2
, (2)
where the operator is called the Mohand transform operator.
Sathya and Rajeswari [32] applied Mohand transform for solving linear partial integro-differential equations. Application of
Mohand transform for solving linear Volterra integro-differential equations was given by Kumar et al. [33]. Aggarwal and
Chaudhary [34] gave a comparative study of Mohand and Laplace transforms. Aggarwal et al. [35] defined Mohand transform of
Bessel’s functions. A comparative study of Mohand and Kamal transforms was given by Aggarwal et al. [36]. Aggarwal et al.
[37] applied Mohand transform and solve the problems of population growth and decay. Aggarwal et al. [38] used Mohand
transform for solving linear Volterra integral equations of second kind. Aggarwal et al. [39] gave a comparative study of Mohand
and Elzaki transforms. A comparative study of Mohand and Aboodh transforms was given by Aggarwal and Chauhan [40]. A
comparative study of Mohand and Sumudu transforms was given by Aggarwal and Sharma [41].
In this paper, we are giving a new application of Mohand transform for handling Abel’s integral equation and explain all
procedure by giving some numerical applications in application section.