##
**Solving Abel’s type integral equation with Mikusinski’s operator of fractional order.**
*(English)*
Zbl 1269.45003

Summary: This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

44A40 | Calculus of Mikusiński and other operational calculi |

### Keywords:

integral equation of Abel’s type; Mikusinski’s operational calculus; Mikusinski’s operator of fractional order
PDF
BibTeX
XML
Cite

\textit{M. Li} and \textit{W. Zhao}, Adv. Math. Phys. 2013, Article ID 806984, 4 p. (2013; Zbl 1269.45003)

Full Text:
DOI

### References:

[1] | N. H. Abel, “Solution de quelques problèmes à l’aide d’intégrales définies,” Magaziu for Naturvidenskaberue, Alu-gang I, Bînd 2, Christiania, pp. 11-18, 1823. |

[2] | R. Gorenflo and S. Vessella, Abel Integral Equations, Springer, 1991. · Zbl 0883.65114 |

[3] | P. P. B. Eggermont, “On Galerkin methods for Abel-type integral equations,” SIAM Journal on Numerical Analysis, vol. 25, no. 5, pp. 1093-1117, 1988. · Zbl 0665.65096 |

[4] | A. Chakrabarti and A. J. George, “Diagonalizable generalized Abel integral operators,” SIAM Journal on Applied Mathematics, vol. 57, no. 2, pp. 568-575, 1997. · Zbl 0874.45001 |

[5] | J. R. Hatcher, “A nonlinear boundary problem,” Proceedings of the American Mathematical Society, vol. 95, no. 3, pp. 441-448, 1985. · Zbl 0591.30038 |

[6] | G. N. Minerbo and M. E. Levy, “Inversion of Abel’s integral equation by means of orthogonal polynomials,” SIAM Journal on Numerical Analysis, vol. 6, no. 4, pp. 598-616, 1969. · Zbl 0213.17102 |

[7] | J. D. Tamarkin, “On integrable solutions of Abel’s integral equation,” The Annals of Mathematics, vol. 31, no. 2, pp. 219-229, 1930. · JFM 56.0347.02 |

[8] | D. B. Sumner, “Abel’s integral equation as a convolution transform,” Proceedings of the American Mathematical Society, vol. 7, no. 1, pp. 82-86, 1956. · Zbl 0070.10303 |

[9] | I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York. NY, USA, 1964. |

[10] | C. E. Smith, “A theorem of Abel and its application to the development of a function in terms of Bessel’s functions,” Transactions of the American Mathematical Society, vol. 8, no. 1, pp. 92-106, 1907. · JFM 38.0490.01 |

[11] | S. Sohrabi, “Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation,” Ain Shams Engineering Journal, vol. 2, no. 3-4, pp. 249-254, 2011. |

[12] | A. Antoniadis, J. Q. Fan, and I. Gijbels, “A wavelet method for unfolding sphere size distributions,” The Canadian Journal of Statistics, vol. 29, no. 2, pp. 251-268, 2001. · Zbl 0974.62029 |

[13] | R. J. Hughes, “Semigroups of unbounded linear operators in Banach space,” Transactions of the American Mathematical Society, vol. 230, pp. 113-145, 1977. · Zbl 0359.47020 |

[14] | K. Ito and J. Turi, “Numerical methods for a class of singular integro-differential equations based on semigroup approximation,” SIAM Journal on Numerical Analysis, vol. 28, no. 6, pp. 1698-1722, 1991. · Zbl 0744.65103 |

[15] | O. P. Singh, V. K. Singh, and R. K. Pandey, “A stable numerical inversion of Abel’s integral equation using almost Bernstein operational matrix,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 111, no. 1, pp. 245-252, 2010. |

[16] | R. K. Pandey, O. P. Singh, and V. K. Singh, “Efficient algorithms to solve singular integral equations of Abel type,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 664-676, 2009. · Zbl 1165.45303 |

[17] | M. Khan and M. A. Gondal, “A reliable treatment of Abel’s second kind singular integral equations,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1666-1670, 2012. · Zbl 1253.65202 |

[18] | R. Weiss, “Product integration for the generalized Abel equation,” Mathematics of Computation, vol. 26, pp. 177-190, 1972. · Zbl 0257.45015 |

[19] | W. C. Brenke, “An application of Abel’s integral equation,” The American Mathematical Monthly, vol. 29, no. 2, pp. 58-60, 1922. · JFM 48.0491.03 |

[20] | E. B. Hansen, “On drying of laundry,” SIAM Journal on Applied Mathematics, vol. 52, no. 5, pp. 1360-1369, 1992. · Zbl 0753.35128 |

[21] | A. T. Lonseth, “Sources and applications of integral equations,” SIAM Review, vol. 19, no. 2, pp. 241-278, 1977. · Zbl 0363.45001 |

[22] | Y. H. Jang, “Distribution of three-dimensional islands from two-dimensional line segment length distribution,” Wear, vol. 257, no. 1-2, pp. 131-137, 2004. |

[23] | L. Bougoffa, R. C. Rach, and A. Mennouni, “A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1785-1793, 2011. · Zbl 1252.65207 |

[24] | J. Mikusiński, Operational Calculus, Pergamon Press, 1959. · Zbl 0088.33002 |

[25] | T. K. Boehme, “The convolution integral,” SIAM Review, vol. 10, no. 4, pp. 407-416, 1968. · Zbl 0179.16902 |

[26] | G. Bengochea and L. Verde-Star, “Linear algebraic foundations of the operational calculi,” Advances in Applied Mathematics, vol. 47, no. 2, pp. 330-351, 2011. · Zbl 1223.44003 |

[27] | V. I. Istr\ua\ctescu, Introduction to Linear Operator Theory, vol. 65, Marcel Dekker, New York, NY, USA, 1981. · Zbl 0457.47001 |

[28] | M. Li and W. Zhao, Analysis of Min-Plus Algebra, Nova Science Publishers, 2011. |

[29] | A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, Fla, USA, 1998. · Zbl 0896.45001 |

[30] | http://eqworld.ipmnet.ru/. |

[31] | C. Cattani and A. Kudreyko, “Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4164-4171, 2010. · Zbl 1186.65160 |

[32] | C. Cattani, M. Scalia, E. Laserra, I. Bochicchio, and K. K. Nandi, “Correct light deflection in Weyl conformal gravity,” Physical Review D, vol. 87, no. 4, Article ID 47503, 4 pages, 2013. |

[33] | C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012. · Zbl 1264.42016 |

[34] | M. Carlini, T. Honorati, and S. Castellucci, “Photovoltaic greenhouses: comparison of optical and thermal behaviour for energy savings,” Mathematical Problems in Engineering, vol. 2012, Article ID 743764, 10 pages, 2012. · Zbl 06173540 |

[35] | M. Carlini and S. Castellucci, “Modelling the vertical heat exchanger in thermal basin,” in Computational Science and Its Applications, vol. 6785 of Lecture Notes in Computer Science, pp. 277-286, Springer, 2011. |

[36] | M. Carlini, C. Cattani, and A. Tucci, “Optical modelling of square solar concentrator,” in Computational Science and Its Applications, vol. 6785 of Lecture Notes in Computer Science, pp. 287-295, Springer, 2011. |

[37] | E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of Quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. · Zbl 1191.35220 |

[38] | E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011. · Zbl 1248.37075 |

[39] | C. Toma, “Advanced signal processing and command synthesis for memory-limited complex systems,” Mathematical Problems in Engineering, vol. 2012, Article ID 927821, 13 pages, 2012. · Zbl 1264.94071 |

[40] | G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems in Engineering, vol. 2010, Article ID 324818, 11 pages, 2010. · Zbl 1191.37052 |

[41] | J. Yang, Y. Chen, and M. Scalia, “Construction of affine invariant functions in spatial domain,” Mathematical Problems in Engineering, vol. 2012, Article ID 690262, 11 pages, 2012. · Zbl 1264.94031 |

[42] | J. W. Yang, Z. R. Chen, W.-S. Chen, and Y. J. Chen, “Robust affine invariant descriptors,” Mathematical Problems in Engineering, vol. 2011, Article ID 185303, 15 pages, 2011. · Zbl 1213.68555 |

[43] | Z. Jiao, Y.-Q. Chen, and I. Podlubny, “Distributed-Order Dynamic Systems,” Springer, 2011. |

[44] | H. Sheng, Y.-Q. Chen, and T.-S. Qiu, Fractional Processes and Fractional Order Signal Processing, Springer, 2012. |

[45] | H. G. Sun, Y.-Q. Chen, and W. Chen, “Random-order fractional differential equation models,” Signal Processing, vol. 91, no. 3, pp. 525-530, 2011. · Zbl 1203.94056 |

[46] | S. V. Muniandy, W. X. Chew, and C. S. Wong, “Fractional dynamics in the light scattering intensity fluctuation in dusty plasma,” Physics of Plasmas, vol. 18, no. 1, Article ID 013701, 8 pages, 2011. |

[47] | H. Asgari, S. V. Muniandy, and C. S. Wong, “Stochastic dynamics of charge fluctuations in dusty plasma: a non-Markovian approach,” Physics of Plasmas, vol. 18, no. 8, Article ID 083709, 4 pages, 2011. · Zbl 1283.76061 |

[48] | C. H. Eab and S. C. Lim, “Accelerating and retarding anomalous diffusion,” Journal of Physics A, vol. 45, no. 14, Article ID 145001, 17 pages, 2012. · Zbl 1237.35166 |

[49] | C. H. Eab and S. C. Lim, “Fractional Langevin equations of distributed order,” Physical Review E, vol. 83, no. 3, Article ID 031136, 10 pages, 2011. |

[50] | L.-T. Ko, J.-E. Chen, Y.-S. Shieh, H.-C. Hsin, and T.-Y. Sung, “Difference-equation-based digital frequency synthesizer,” Mathematical Problems in Engineering, vol. 2012, Article ID 784270, 12 pages, 2012. · Zbl 06173572 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.