A Generalized Barycentric Rational Interpolation Method for Generalized Abel Integral Equations
- PDF / 733,715 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 101 Downloads / 190 Views
A Generalized Barycentric Rational Interpolation Method for Generalized Abel Integral Equations H. Azin1 · F. Mohammadi1
· D. Baleanu2,3
Accepted: 31 August 2020 © Springer Nature India Private Limited 2020
Abstract The paper is devoted to the numerical solution of generalized Abel integral equation. First, the generalized barycentric rational interpolants have been introduced and their properties investigated thoroughly. Then, a numerical method based on these barycentric rational interpolations and the Legendre–Gauss quadrature rule is developed for solving the generalized Abel integral equation. The main advantages of the presented method is that it provides an infinitely smooth approximate solution with no real poles for the generalized Abel integral equation. Keywords Generalized Abel integral equation · Generalized barycentric rational interpolation · Legendre–Gauss quadrature · Error analysis Mathematics Subject Classification 45E10 · 45D05 · 41A20
Introduction Integral equations are special type of functional equations which contain integral operators. Many problems in the fields of applied mathematics, physics, chemistry, fluid mechanics, biological systems, quantum mechanics, metallurgy and scattering theory can be formulated in the form of Integral equations [1,2]. Recently, several numerical methods such as wavelet method [3–7], direct quadrature method [8], reproducing kernel method [9], radial basis functions [10], collocation method [11], meshless local discrete Galerkin [12], multistep collocation method [13], Tau approximate solution [14] have been applied to approximate solution of integral equations.
B
F. Mohammadi [email protected] D. Baleanu [email protected]
1
Department of Mathematics, Faculty of Science, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran
2
Department of Mathematics, Cankaya University, Balgat, 06530 Ankara, Turkey
3
Institute of Space Sciences, Magurele-Bucharest, Romania 0123456789().: V,-vol
123
140
Page 2 of 12
Int. J. Appl. Comput. Math
(2020) 6:140
In 1823, the Norwegian mathematician Niels Henrik Abel studied the following mechanical problem: In the two-dimensional x y-plane find a curve x = φ(y) with a property that, the total time of descent for a particle sliding on it only due to gravity g, must be equal to a prescribed function T (y) of the initial height y. In absence of friction and by the principle of conservation of energy, Abel’s solution to this problem were reduced to that of solving the following integral equation y u(t) (1) dt = 2gT (y), √ y−t 0 where u(t) =
1+
dφ dt
√
2
. Later, Abel studied the general form of the above integral
equations by replacing y − t by (y − t)ν , 0 < ν < 1. Accordingly, in honour of Abel, the following forms of Volterra integral equations are considered as generalized Abel integral equation (GAIE): x u(t) dt = f (x), 0 < ν < 1 a > 0. (2) ν a (x − t) x u(t) dt = f (x), 0 < ν < 1 a > 0. (3) u(x) + (x − t)ν a The GAIE, have enormous direct applications in scattering theor
Data Loading...
