Evaluation of inverse Fourier transforms through romberg integration

Abstract

a b s t r a c tThis study deals with evaluation of inverse integral transforms through numerical integration. Arguments are presented to explain that such a numerical evaluation is mathematically incorrect. In recent years, a sudden increase has been noticed in the number of research publications in which Romberg integration is said to have used in evaluating the inverse Fourier transforms. In this regard, even the references cited in these papers do not support such an evaluation but suggest contrary. An immediate concern is to caution the researchers studying the source problems in the field of solid mechanics over the use and approval of an erroneous numerical technique.

Numerical evaluation of inverse integral transforms: Dynamic response of elastic materials

“…The reference of Press et al (1992) is cited in support of this choice. But, it has been explained very clearly in Sharma (2010a) that Romberg integration cannot be used for such integrals and the cited reference (Press et al, 1992) never supported this adventure. Moreover, on the evaluation of such integrals, a relevant reference (Ewing et al, 1957) suggests to use the contour integration rather than numerical integration (Sharma, 2010a).…”

“…However, none of these researchers ever discussed the branch points or the singularities present in the corresponding integrands. Sharma (2010a) has discussed the errors deliberately committed by these researchers while going for this numerical adventure. The relevant studies mainly consider calculating the response of elastic materials to various sources of deformation.…”

“…The relevant studies mainly consider calculating the response of elastic materials to various sources of deformation. Nearly fifty of such papers are listed in Sharma (2010a), which have appeared in reputed international journals devoted to mechanics of solids. Irony is that even after this publication (Sharma, 2010a), these authors did not stop their erroneous numerical adventure on inverse Fourier transforms.…”

“…Nearly fifty of such papers are listed in Sharma (2010a), which have appeared in reputed international journals devoted to mechanics of solids. Irony is that even after this publication (Sharma, 2010a), these authors did not stop their erroneous numerical adventure on inverse Fourier transforms. In most of these papers, the inverse Laplace transforms are also calculated numerically.…”

Numerical modeling of thermoelastic generation of ultrasound by laser irradiation in the coupled thermoelasticity

“…This approach was applied to investigate epicentral waveforms in [3,7,8] , the generation process by line focused sources [9] or the influence of the hyperbolic term in the heat conduction equation [10] . Simplifications in the numerical inversion techniques are, however, controversial [11] . An alternative way to solve this coupled problem is shown in [12,13] using the reciprocity theorem with applications to a half space and plates.…”