Non-classical sinc collocation method for approximating Hallen's integral equation

Document Type : Research Article

Authors

Department of Mathematics, Faculty of Sciences, University of Kurdistan, Sanandaj, Kurdistan, Iran

Abstract

In this paper, we propose a novel numerical method for solving Hallen’s integral equation, based on the sinc collocation approximation. The key innovation of our approach lies in the incorporation of weight functions into the traditional sinc-expansion framework. By leveraging the properties of sinc collocation, we transform Hallen’s integral equation into a system of algebraic equations, which can be solved efficiently.  Our method involves discretizing the singular kernel of Hallen’s integral equation and then applying the sinc approximation. Additionally, we provide a detailed analysis of the convergence and error estimation of the proposed method. Numerical results are presented for three distinct values of $\lambda$ and $l$, as well as for three different weight functions: $w(t)=1+\sin(\pi t)$, $w(t)=1+\cos(\frac{\pi t}{2})$ and $w(t)=1+t$.

Keywords

Main Subjects


[1] A. Alipanah, Nonclassical pseudospectral method for a class of singular boundary value problems
arising in physiology, Appl. Math. 2 (2012) 1–4.
[2] A. Alipanah, Solution of Hallens integral equation by using radial basis functions, Math. Rep. 15
(2013) 211–220.
[3] A. Alipanah, K. Mohammadi, M. Ghasemi, Numerical solution of third-Order boundary value
problems using non-classical sinc-collocation method, Comput. Meth. Diff. Eq. 11 (2023) 100403.
[4] A. Alipanah, K. Mohammadi, R.M. Haji, Numerical solution of singularly perturbed singular third
order boundary value problems with nonclassical sinc method, Comput. Meth. Diff. Eq. 22 (2024)
100459.
[5] M. Bayjja, M. Boussouis, N.A. Touhami, K. Zeljami, Comparison between solution of Pocklingtons
and Hallens integral equations for thin wire antennas using method of moments and Haar wavelet,
Comput. Meth. Diff. Eq. 12 (2015) 931–942.
[6] A. Eftekhari, Spectral poly-sinc collocation method for solving a singular nonlinear BVP of
reaction-diffusion with Michaelis-Menten kinetics in a catalyst/biocatalyst, Iranian J. Math. Chem.
14 (2023) 77–96.
[7] A. Eftekhari, DE sinc-collocation method for solving a class of second-order nonlinear BVPs,
Math. Interdisc. Res. 6 (2021) 11–22.
[8] M. El-Gamel, S. Behiry, H. Hashish, Numerical method for the solution of special nonlinear fourth-
order boundary value problems, Appl. Math. Comput. 145 (2003) 717–734.
[9] G. Fikioris, T.T. Wu, On the application of numerical methods to Hallen’s equation, IEEE T. An-
tenn. Propag. 49 (2001) 383–392.
[10] E. Hallen, Theoretical Investigation into the Transmitting and Receiving Qualites of Antennas,
Nova Acta Upssala, 1938.
[11] E. Hallen, Exact treatment of antenna current wave reflection at the end of a tube-shaped cylindri-
cal antenna, IEEE T. Antenn. Propag. 4 (1956) 479–491.
[12] S.-J. Lai, B.-Z. Wang, Y. Duan, Meshless radial basis functions method for solving Hallens Integral
equation, Appl. Comput. Electromagn. Soc. J. 27 (2012) 9–13.
[13] J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, 1992.
[14] K. Mohammadi, A. Alipanah, Numerical solution of the system of second-order integro-differential
equations using non-classical double sinc method, Res. Appl. Math. 19 (2023) 100381.
[15] K. Mohammadi,A. Alipanah, A non-classical sinc-collocation method for the solution of singular
boundary value problems arising in physiology, Int. J. Comput. Math. 99 (2022) 1941–1967.
[16] D. Poljak, V. Roje, Finite element technique for solving time-domain Hallen integral equation,
Tenth International Conference on Antennas and Propagation (Conf. Publ. No. 436). 1 (1997) 225–
228.
[17] A. Saadatmandi, M. Razzaghi, M. Dehghan, Sinc-collocation methods for the solution of Hallen’s
integral equation, J. Electromagn. Waves Appl. 19 (2005) 245–256.
[18] A. Saadatmandi, A. Khani, M.R. Azizi, Numerical calculation of fractional derivatives for the sinc
functions via Legendre polynomials, Math. Interdisc. Res. 5 (2020) 71–86.
[19] A. Saadatmandi, A. Khani, M.R. Azizi, Solution of Hallen’s integral equation using multiwavelets,
Comput. Phys. Comm. 168 (2005) 187–197.
[20] B. Shizgal, A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and
related problems, Comput. Phys. Comm. 41 (1981) 309–328.
[21] B.D. Shizgal, H. Chen, The quadrature discretization method (QDM) in the solution of the Schr¨o
dinger equation with nonclassical basis functions, J. Chem. Phys. 104 (1996) 4137–4150.
[22] F. Stenger, Approximations via Whittaker’s cardinal function, J. Approx. Theory. 17 (1976) 222–
240.
[23] F. Stenger, A Sinc-Galerkin method of solution of boundary value problems, Math. Comput. 33
(1979) 85–109.
[24] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer 2012.
[25] T.T.Wu, Theory of the dipole antenna and the two-wire transmission line, J. Math. Phys. 2 (1961)
550–574.