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Generalized quadrature for solving singular integral equations of Abel type in application to infrared tomography. (English) Zbl 1416.65552
Summary: We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytic computation of the singular integral. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it makes sense to apply regularization in case of highly noisy (about 10%) sources only. That is due to the known fact that Volterra equations of the first kind enjoy selfregularization property.

MSC:
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
92C55 Biomedical imaging and signal processing
Software:
EDA
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[1] Ă…kesson, E. O.; Daun, K. J., Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization, Appl. Opt., 47, 3, 407-416, (2008)
[2] Anderssen, R. S.; de Hoog, F. R., Abel integral equations, (Goldberg, M. A., Numerical Solution of Integral Equations, (1990), Plenum Press), 373-410 · Zbl 0729.65111
[3] Belotserkovskii, S. M.; Lifanov, I. K., Method of discrete vortices, (1993), CRC Press
[4] Boikov, I. V.; Kudryashova, N. Yu., Approximate methods for singular integral equations in exceptional cases, Differ. Equ., 36, 9, 1360-1369, (2000) · Zbl 0991.65143
[5] Brunner, H.; van der Houwen, P. J., The numerical solution of Volterra equations, (1986), North-Holland · Zbl 0611.65092
[6] Dasch, C. J., One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods, Appl. Opt., 31, 8, 1146-1152, (1992)
[7] Daun, K. J., Infrared species limited data tomography through Tikhonov reconstruction, J. Quant. Spectrosc. Radiat. Transf., 111, 1, 105-115, (2010)
[8] Daun, K. J.; Thomson, K. A.; Liu, F.; Smallwood, G. J., Deconvolution of axisymmetric flame properties using Tikhonov regularization, Appl. Opt., 45, 19, 4638-4646, (2006)
[9] Deutsch, M.; Beniaminy, I., Derivative-free inversion of Abel’s integral equation, Appl. Phys. Lett., 41, 1, 27-28, (1982)
[10] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Academic Publ. · Zbl 0859.65054
[11] Evseev, V., Optical tomography in combustion, (2012), DTU Chemical Eng., PhD Thesis
[12] V. Evseev, A. Fateev, V. Sizikov, S. Clausen, K.L. Nielsen, On the development of methods and equipment for 2D-tomography in combustion, Report on Annual Meeting of Danish Physical Society, 21-22 June 2011, 32 pp.
[13] Ghanbari, M.; Askaripour, M.; Khezrimotlagh, D., Numerical solution of singular integral equations using Haar wavelet, Aust. J. Basic Appl. Sci., 4, 12, 5852-5855, (2010)
[14] Gorenflo, R.; Vessella, S., Abel integral equations, (1991), Springer · Zbl 0717.45002
[15] Hall, R. J.; Bonczyk, P. A., Sooting flame thermometry using emission/absorption tomography, Appl. Opt., 29, 31, 4590-4598, (1990)
[16] Hansen, P. C., Discrete inverse problems: insight and algorithms, (2010), SIAM Publ. · Zbl 1197.65054
[17] Korn, G. A.; Korn, T. M., Mathematical handbook for scientists and engineers, (1961), MGraw-Hill Book Company · Zbl 0121.00103
[18] Kosarev, E. L., The numerical solution of Abel’s integral equation, Comput. Math. Math. Phys., 13, 6, 271-277, (1973) · Zbl 0294.65067
[19] Krylov, V. I., Approximate calculation of integrals, (2005), Dover Publications · Zbl 1152.65005
[20] Krylov, V. I.; Bobkov, V. V.; Monastyrnyi, P. I., Computational methods, vol. 2, (1977), Nauka
[21] Li, M.; Zhao, W., Solving Abel’s type integral equation with Mikusinski’s operator of fractional order, Adv. Math. Phys., (2013), 4 pp · Zbl 1269.45003
[22] Mandal, B. N.; Chakrabarti, A., Applied singular integral equations, (2011), CRC Press · Zbl 1234.45002
[23] Martinez, W. L.; Martinez, A. R.; Solka, J. L., Exploratory data analysis with MATLAB, (2010), CRC Press · Zbl 1270.62014
[24] Minerbo, G. N.; Levy, M. E., Inversion on Abel’s integral equation by means of orthogonal polynomials, SIAM J. Numer. Anal., 6, 4, 598-616, (1969) · Zbl 0213.17102
[25] Morozov, V. A., Methods for solving incorrectly posed problems, (1984), Springer
[26] Natterer, F., The mathematics of computerized tomography, (1986), Wiley · Zbl 0617.92001
[27] Pandey, R. K.; Singh, O. P.; Singh, V. K., Efficient algorithms to solve singular integral equations of Abel type, Comput. Math. Appl., 57, 4, 664-676, (2009) · Zbl 1165.45303
[28] Pikalov, V. V.; Preobrazhenskii, N. G., Computer-aided tomography and physical experiment, Sov. Phys. Usp., 26, 11, 974-990, (1983)
[29] Porter, R. W., Numerical solution for local coefficients in axisymmetric self-absorbed sources, SIAM Rev., 6, 3, 228-242, (1964)
[30] Preobrazhensky, N. G.; Pikalov, V. V., Unstable problems of plasma diagnostics, (1982), Nauka
[31] Saadamandi, A.; Dehghan, M., A collocation method for solving Abel’s integral equations of first and second kinds, Z. Naturforsch., 63(a), 752-756, (2008)
[32] Sidorov, D., Integral dynamical models: singularities, signals and control, (2014), World Sci. Publ.
[33] Sidorov, D.; Tynda, A.; Muftahov, I., Numerical solution of weakly regular Volterra integral equations of the first kind, (2014) · Zbl 1308.65222
[34] Singh, V. K.; Pandey, R. K.; Singh, O. P., New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials, Appl. Math. Sci., 3, 5, 241-255, (2009) · Zbl 1175.65152
[35] Sizikov, V. S., Mathematical methods for processing the results of measurements, (2001), Politekhnika
[36] Sizikov, V. S., Infrared tomography of hot gas: mathematical model of active-passive diagnostics, Sci. Tech. J. ITMO Univ., 6, 88, 1-17, (2013)
[37] Sizikov, V. S.; Smirnov, A. V.; Fedorov, B. A., Numerical solution of the abelian singular integral equation by the generalized quadrature method, Rus. Math. (Iz. VUZ), 48, 8, 59-66, (2004) · Zbl 1100.65128
[38] Tikhonov, A. N.; Arsenin, V. Ya., Solutions of ill-posed problems, (1977), Wiley
[39] Verlan’, A. F.; Sizikov, V. S., Integral equations: methods, algorithms, programs, (1986), Nauk. Dumka · Zbl 0635.65134
[40] Voskoboynikov, Yu. E.; Preobrazhensky, N. G.; Sedel’nikov, A. I., Mathematical treatment of experiment in molecular gas dynamics, (1984), Nauka
[41] Wazwaz, A. M., Linear and nonlinear integral equations: methods and applications, (2011), Springer
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