Commun. Comput. Phys., 8 (2010), pp. 351-373. |
A New Stable Algorithm to Compute Hankel Transform Using Chebyshev Wavelets Rajesh K. Pandey ^{1}, Vineet K. Singh ^{2}, Om P. Singh ^{3*} 1 PDPM Indian Institute of Information Technology Design and Manufacturing, Jabalpur, India.2 Birla Institute of Technology and Science-Pilani, Goa Campus, Goa, India. 3 Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi, India. Received 5 June 2009; Accepted (in revised version) 21 December 2009 Available online 12 March 2010 doi:10.4208/cicp.050609.211209a Abstract A new stable numerical method, based on Chebyshev wavelets for numerical evaluation of Hankel transform, is proposed in this paper. The Chebyshev wavelets are used as a basis to expand a part of the integrand, $rf(r)$, appearing in the Hankel transform integral. This transforms the Hankel transform integral into a Fourier-Bessel series. By truncating the series, an efficient and stable algorithm is obtained for the numerical evaluations of the Hankel transforms of order $\nu > -1$. The method is quite accurate and stable, as illustrated by given numerical examples with varying degree of random noise terms $\varepsilon \theta_i$ added to the data function $f(r)$, where $\theta_i$ is a uniform random variable with values in $[-1,1]$. Finally, an application of the proposed method is given for solving the heat equation in an infinite cylinder with a radiation condition. AMS subject classifications: 65R10 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Hankel transforms, Bessel functions, Chebyshev wavelets, random noise term. *Corresponding author. Email: wavelet_r@yahoo.co.in (R. K. Pandey), singhom@gmail.com (O. P. Singh), vks1itbhu@gmail.com (V. K. Singh) |