Solution of the Generalized Abel Integral Equation by Using Almost Bernstein Operational Matrix

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DOI: 10.4236/ajcm.2011.14026    4,365 Downloads   8,980 Views   Citations


A direct almost Bernstein operational matrix of integration is used to propose a stable algorithm for numerical inversion of the generalized Abel integral equation. The applicability of the earlier proposed methods was restricted to the numerical inversion of a part of the generalized Abel integral equation. The method is quite accurate and stable as illustrated by applying it to intensity data with and without random noise to invert and compare it with the known analytical inverse. Thus it is a good method for applying to experimental intensities distorted by noise.

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S. Dixit, R. Pandey, S. Kumar and O. Singh, "Solution of the Generalized Abel Integral Equation by Using Almost Bernstein Operational Matrix," American Journal of Computational Mathematics, Vol. 1 No. 4, 2011, pp. 226-234. doi: 10.4236/ajcm.2011.14026.


[1] N. H. Abel, “Resolution d’un Probleme de Mechanique,” Journal für die Reine und Angewandte Mathematik, Vol. 1826, No. 1, 2009, pp. 153-157. doi:10.1515/crll.1826.1.153
[2] A. J. Jakeman and R. S. Anderssen, “Abel Type Integral Equations in Stereology: I. General Discussion,” Journal of Microscopy, Vol. 105, No. 2, 1975, pp. 121-133. doi:10.1111/j.1365-2818.1975.tb04045.x
[3] J. B. Macelwane, “Evidence on the Interior of the Earth Derived from Seismic Sources”, In: B. Gutenberg, Ed., Internal Constitution of the Earth, Dover, New York, 1951, pp. 227-304.
[4] S. B. Healy, J. Haase and O. Lesne, “Abel Transform Inversion of Radio Occultation Measurements Made with a Receiver inside the Earth’s Atmosphere,” Annals of Geophysicae, Vol. 20, No. 8, 2002, pp. 1253-1256. doi:10.5194/angeo-20-1253-2002
[5] R. N. Bracewell and A. C. Riddle, “Inversion of Fan-Beam Scans in Radio Astronomy,” Astrophysical Journal, Vol. 150, 1967, pp. 427-434. doi:10.1086/149346
[6] S. C. Solomon, P. B. Hays and V. J. Abreu, “Tomogra- phic Inversion of Satellite Photometry,” Applied Optics, Vol. 23, No. 19, 1984, pp. 3409-3414. doi:10.1364/AO.23.003409
[7] E. L. Kosarev, “Applications of Integral Equations of the First Kind in Experiment Physics,” Compututer Physics Communications, Vol. 20, No. 1, 1980, pp. 69-75. doi:10.1016/0010-4655(80)90110-1
[8] U. Buck, “Inversion of Molecular Scattering Data,” Re- views of Modern Physics, Vol. 46, No. 2, 1974, pp. 369- 389. doi:10.1103/RevModPhys.46.369
[9] H. Hellsten and L. E. Andersson, “An Inverse Method for the Processing of Synthetic Aperture Radar Data,” Inverse Problems, Vol. 3, No. 1, 1987, pp. 111-124. doi:10.1088/0266-5611/3/1/013
[10] R. S. Anderssen and R. B. Calligaro, “Non Destructive Testing of Optical-Fiber Performs,” The Journal of the Australian Mathematical Society. Series B. Applied Ma- thematics, Vol. 23, No. 2, 1981, pp. 127-135. doi:10.1017/S0334270000000138
[11] K. Taketura, “Index Profile Determination of Single-Mode Fiber by the Phase Contrast Method: A Proposed Tech- nique,” Applied Optics, Vol. 21, No. 23, 1982, pp. 4260- 4263. doi:10.1364/AO.21.004260
[12] W. J. Glantschnig, “How Accurately Can One Reconstruct an Index Profile from Transverse Measurement Data,” Journal of Lightwave Technology, Vol. 3, No. 3, 1985, pp. 678-683. doi:10.1109/JLT.1985.1074221
[13] E. Keren, E. Bar-Ziv, I. Glatt and O. Kafri, “Measurements of Temperature Distribution of Flames by Moiré Deflec- tometry,” Applied Optics, Vol. 20, No. 24, 1981, pp. 4263- 4266. doi:10.1364/AO.20.004263
[14] C. J. Tallents, M. D. J. Burgess and B. Luther-Davies, “The Determination of Electron Density Profiles from Re- fraction Measurements Obtained Using Holographic In- terferometry,” Optics Communications, Vol. 44, No. 6, 1983, pp. 384-387. doi:10.1016/0030-4018(83)90222-5
[15] C. Fleurier and J. Chapelle, “Inversion of Abel’s Integral Equation Application to Plasma Spectroscopy,” Compu- tuter Physics Communications, Vol. 7, No. 4, 1974, pp. 200-206. doi:10.1016/0010-4655(74)90089-7
[16] K. M. Hanson, “Tomographic Reconstruction of Axially Symmetric Objects from a Single Radiograph,” Proceed- ings of SPIE 16th International Conference on High Speed Photography and Photonics, Vol. 491, 1984, pp. 180-187.
[17] W. J. Glantschnig and A. Holliday, “Mass Fraction Profiling Based on X-Ray Tomography and Its Application to Characterising Porous Silica Boules,” Applied Optics, Vol. 26, No. 6, 1987, pp. 983-989. doi:10.1364/AO.26.000983
[18] A. M. Cormack, “Representation of a function by its line integrals with some radiological applications”, Journal of Applied Physics, Vol. 34, No. 9, 1963, pp. 2722-2727. doi:10.1063/1.1729798
[19] M. Deutsch, A. Notea and D. Pal, “Inversion of Abel’s Integral Equation and Its Application to NDT by X-Ray Radiography”, NDT. International, Vol. 23, No.1, 1990, pp. 32-38.
[20] A. Chakrabarti, “Solution of the Generalized Abel Integral Equation”, Journal of Integral Equations and Applications, Vol. 20, No. 1, 2008, pp. 1-11. doi:10.1216/JIE-2008-20-1-1
[21] F. D. Gakhov, “Boundary Value Problems,” Pergamon Press, Oxford, 1966.

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