Generalized Abel Inversion Using Homotopy Perturbation Method

Sunil Kumar, Om P. Singh, Sandeep Dixit

.

DOI: 10.4236/am.2011.22029                

Abstract

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. Obtaining the physically relevant quantity from the measured one requires, therefore the inversion of the Abel’s integral equation. The aim of this letter is to present a user friendly algorithm to invert generalized Abel integral equation by using homotopy perturbation method. The stability of the algorithm is analysed. The validity and applicability of this powerful technique is illustrated through various particular cases which demonstrate its efficiency and simplicity in solving these types of integral equations.

Keywords

Generalized Abel Integral Equation, Homotopy Perturbation Method, Noise Term, Stability

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. H. Abel, “Resolution d’un Probleme,” Journal fur die reine und angewandte Mathematik, Vol. 1, 1826, pp. 153-157. doi:10.1515/crll.1826.1.153
[2]	L. Mach, Wiener Akademie Berlin fur Mathematik und Physik Klasse, Vol. 105, 1896, p. 605.
[3]	S. B. Healy, J. Haase and O. Lense, “Abel Transform Inversion of Radio Occultation Measurement Made with a Receiver inside the Earth’s Atmosphere,” Annales Geophysicae, Vol. 20, 2002, p. 1253. doi:10.5194/angeo-20-1253-2002
[4]	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
[5]	S. C. Soloman, P. B. Hays and V. J. Abreu, “Tomographic Inversion of Satellite Photometry,” Applied Optics, Vol. 23, 1984, pp. 3409-3414. doi:10.1364/AO.23.003409
[6]	E. L. Kosarev, “Applications of Integral Equations of the First Kind in Experiment Physics,” Computer Physics Communication, Vol. 20, 1980, pp. 69-75. doi:10.1016/0010-4655(80)90110-1
[7]	C. J. Tallents, M. D. J. Burgess and B. Luther-Davies, “The Determination of Electron Density Profile from Refraction Measurements Obtained Using Holographic Interferometry,” Optics Communications, Vol. 4, 1983, pp. 384-387. doi:10.1016/0030-4018(83)90222-5
[8]	L. J. M. Ignjatovic and A. A. Mihajlov, “The Realization of Abel’s Inversion in the Case of Discharge with Undetermined Radius,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 72, 2002, pp. 677-689. doi:10.1016/S0022-4073(01)00149-2
[9]	M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek and M. L. Brake, “Abel Inversion Applied to Experimental Spectroscopic Data with Off Axis Peaks,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 55, No. 2, 1996, pp. 231-243. doi:10.1016/0022-4073(95)00149-2
[10]	F. G. Tricomi, “Integral Equations”, Interscience, New York, 1975.
[11]	G. N. Minerbo and M. E. Levy, “Inversion of Abel Integral Equation by Means of Orthogonal Polynomials,” SIAM Journal of Computational Physics, Vol. 6, 1969, pp. 598-616.
[12]	M. Deutsch and I. Beniaminy, “Derivatives Free Inversion of Abel’s Integral Equations,” Applied Physics Letters, Vol. 41, 1982, pp. 27-28. doi:10.1063/1.93309
[13]	M. Deutsch, A. Notea and D. Pal, “Inversion of Abel’s Integral Equations and Its Application to NDT by X-Ray Radiography,” NDT International, Vol. 23, No. 1, 1990, pp. 32-38.
[14]	J. P. Lanquart, “Error Attenuation in Abel Inversion,” Journal of Computational Physics, Vol. 47, 1982, pp. 434-443. doi:10.1016/0021-9991(82)90092-4
[15]	L. J. M. Ignjatovic and A. A. Mihajlov, “The Realization of Abel’s Inversion in the Case of Discharge with Undetermined Radious,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 72, 2002, pp. 677-689. doi:10.1016/S0022-4073(01)00149-2
[16]	V. K. Singh, R. K. Pandey and O. P. Singh, “New Stable Numerical Solution of Singular Integral Equations of Abel Type by Using Normalized Bernstein Polynomials,” Applied Mathematical Sciences, Vol. 3, No. 5, 2009, pp. 241-255.
[17]	O. P. Singh, V. K. Singh and R. K. Pandey, “New Stable Numerical Inversion of Abel Integral Equation Using Almost Bernstein Operational Matrix,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 111, 2010, pp. 245-252. doi:10.1016/j.jqsrt.2009.07.007
[18]	I. Beniaminy and M. Deutsch, “Abel Stable High Accuracy Programme for the Inversion of Abel’s Integral Equation,” Computer Physics Communication, Vol. 27, 1982, pp. 415-422. doi:10.1016/0010-4655(82)90102-3
[19]	D. A. Murio, D. G. Hinestroza and C. E. Mejia, “New Stable Numerical Inversion of Abel’s Integral Equation,” Computers and Mathematics with Applications, Vol. 23, No. 11, 1992, pp. 3-11. doi:10.1016/0898-1221(92)90064-O
[20]	S. Ma, H. Gao, L. Wu and G. Zhang, “Abel Inversion Using Legendre Polynomials Approximations,” Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 109, 2008, pp. 1745-1757. doi:10.1016/j.jqsrt.2008.01.013
[21]	S. Ma, H. Gao, G. Zhang and L. Wu, “Abel Inversion Using Legendre Wavelet Expansion, Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 107, 2007, pp. 61-71. doi:10.1016/j.jqsrt.2007.01.054
[22]	J. H. He, “An Approximate Solution Technique Depending upon an Artificial Parameter,” Communications in Nonlinear Science and Numerical Simulation, Vol. 3, No. 2, 1998, pp. 92-97. doi:10.1016/S1007-5704(98)90070-3
[23]	J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3
[24]	J. H. He, “A Review on Some Recently Developed Nonlinear Analytical Technique,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 1, No. 1, 2000, pp. 51-70.