An Application of Wavelet Technique in Numerical Evaluation of Hankel Transforms

Abstract

Abstract This paper endeavors to formulate a stable and fast algorithm for the first time that is quite accurate and fast for numerical evaluation of the Hankel transform using wavelets which are usually difficult to solve analytically so it is required to obtain the approximate solution. So we have proposed an approach depending on separating the integrand r f ( r ) J ν ( p r ) $rf(r){J_\nu}(pr)$ into two components, the slowly varying components r f ( r ) $rf(r)$ and the rapidly oscillating component J ν ( p r ) ${J_\nu}(pr)$ . Then either r f ( r ) $rf(r)$ is expanded into wavelet series using wavelets orthonormal basis which are first derived and truncating the series at an optimal level or approximating r f ( r ) $rf(r)$ by a quadratic over the subinterval using the Filon quadrature philosophy. A good covenant between the obtained solution and some well-known results has been obtained. The solutions obtained by proposed wavelet method indicate that the approach is easy to implement and computationally very attractive. The novelty of our method is that we give error analysis for wavelet method for the first time in literature.