Mathematical Software for Modified Bessel Functions

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• 作者：Juri Rappoport (17)
  
• 关键词：mathematical software ; modified Bessel functions ; Lanczos tau method ; Chebyshev polynomials ; Kontorovich ; Lebedev integral transforms
• 刊名：Lecture Notes in Computer Science
• 出版年：2014
• 出版时间：2014
• 年：2014
• 卷：8592
• 期：1
• 页码：325-332
• 全文大小：147 KB
• 参考文献：1. Ehrenmark, U.T.: The numerical inversion of two classes of type-small-caps">Kontorovich-Lebedev transform by direct quadrature. J. of Comp. and Appl. Math.?61, 43-2 (1995) CrossRef
  2. Fabijonas, B.R., Lozier, D.W., Rappoport, J.M.: Algorithms and codes for the type-small-caps">Macdonald function: Recent progress and comparisons. Journ. Comput. Appl. Math.?161(1), 179-92 (2003) CrossRef
  3. Gautschi, W.: Computing the type-small-caps">Kontorovich-Lebedev integral transforms and their inverses. BIT Numer. Math.?46, 21-0 (2006) CrossRef
  4. Gil, A., Segura, J., Temme, N.M.: Numerical methods for special functions. SIAM (2007)
  5. Kiyono, T., Murashima, S.: A method of evaluation of the function / K _is( / x). Mem. Fac. Engr. Kyoto Univ.?35(2), 102-27 (1973)
  6. Lebedev, N.N., Skalskaya, I.P.: Some integral transforms related to type-small-caps">Kontorovitch-Lebedev transforms, The questions of the mathematical physics, Leningrad, Nauka, 68-9 (1976) (in Russian)
  7. Luke, Y.L.: Mathematical functions and their approximations. Academic Press, New York (1975)
  8. Olver, F., Lozier, D., Boisvert, R., Clark, C. (eds.): NIST Handbook of mathematical functions. Cambridge University Press (2010)
  9. Rappoport, J.M.: Tables of modified Bessel functions / K _{1/2--em class="a-plus-plus">iτ} ( / x), Nauka, Moscow (1979) (in Russian)
  10. Rappoport, J.M.: The programs and some methods of the Macdonald’s function computation. OVM AN U.S.S.R., Moscow (1991) (in Russian)
  11. Rappoport, J.M.: Some results for modified type-small-caps">Kontorovitch–Lebedev integral transforms. In: Proceedings of the 7th International Colloquium on Finite or Infinite Dimensional Complex Analysis, pp. 473-77. Marcel Dekker Inc (2000)
  12. Rappoport, J.M.: The canonical vector-polynomials at computation of the type-small-caps">Bessel functions of the complex order. Comput. Math. Appl.?41(3/4), 399-06 (2001) CrossRef
  13. Rappoport, J.M.: Some numerical quadrature algorithms for the computation of the type-small-caps">Macdonald function. In: Proceedings of the Third ISAAC Congress. Progress in Analysis, vol. 2, pp. 1223-230. World Scientific Publishing (2003)
  14. Rappoport, J.M.: Dual integral equations for some mixed boundary value problems. In: Proceedings of the 4th ISAAC Congress. Advances in Analysis, pp. 167-76. World Scientific Publishing (2005)
  15. Rappoport, J.M.: The properties, inequalities and numerical approximation of modified type-small-caps">Bessel function. Electronic Transactions on Numerical Analysis?25, 454-66 (2006)
  16. Rappoport, J.M.: Some numerical methods for approximation of modified type-small-caps">Bessel functions. Proc. Appl. Math. Mech.?7(1), 2020017-020018 (2007) CrossRef
  17. Rappoport, J.M.: About modified type-small-caps">Kontorovitch–Lebedev integral transforms and their kernels, Imperial College of Science, Technology and Medicine, Department of Mathematics, London, preprint 09P/001, 31p. (2009)
  18. Rappoport, J.M.: Some integral equations with modified type-small-caps">Bessel function. In: Proceedings of the 5th ISAAC Congress. More Progresses in Analysis, pp. 269-78. World Scientific Publishing (2009)
  19. Rappoport, J.M.: type-small-caps">Chebyshev polynomial approximations for some hypergeometric systems, Isaac Newton Institute of Mathematical Sciences, Cambridge, preprint NI09059-DIS, 8p. (2009)
  20. Temme, N.M.: On the numerical evaluation of the modified bessel function of the third kind. J. of Comp. Phys.?17, 324-37 (1975) CrossRef
  21. Yakubovich, S.B.: Index Transforms. World Scientific Publishing, Singapore (1996) CrossRef
  22. Yakubovich, S.B.: type-small-caps">Beurling’s theorems and inversion formulas for certain index transforms. Opuscula Mathematica?29(1), 93-10 (2009)
  23. Zurina, M.I., Karmazina, L.N.: Tables of modified Bessel functions with imaginary index. / VC AN U.S.S.R., Moscow (1967) (in Russian)
  
• 作者单位：Juri Rappoport (17)
  
  17. Institute for Computer Aided Design, Russian Academy of Sciences, Russia
  
• ISSN：1611-3349

文摘

The high-quality mathematical software for the computation of modified Bessel functions of the second kind with integer, imaginary and complex order and real argument is elaborated. The value of function may be evaluated with high precision for given value of the independent argument x and order r. These codes are addressed to the wide audience of scientists, engineers and technical specialists. The tables of these functions are published. This software improves significantly the capability of computer libraries. These functions arise naturally in boundary value problems involving wedge-shaped or cone-shaped geometries. They are fundamental to mathematical modeling activities in applied science and engineering. Methods of mathematical and numerical analysis are adapted for the creation of appropriate algorithms for these functions, computer codes are written and tested. Power series, Tau method and numerical quadratures of trapezoidal kind are used for the construction of subroutines. New realization of the Lanczos Tau method with minimal residue is proposed for the constructive approximation of the second order differential equations solutions with polynomial coefficients. A Tau method computational scheme is applied for the constructive approximation of a system of differential equations solutions related to the differential equation of hypergeometric type. Various vector perturbations are discussed. Our choice of the perturbation term is a shifted Chebyshev polynomial with a special form of selected transition and normalization. The minimality conditions for the perturbation term are found for one equation. They are sufficiently simple for verification in a number of important cases. Tau method’s approach gives a big advantage in the economy of computer’s time. The mathematical software for kernels of Lebedev type index transforms -modified Bessel functions of the second kind with complex order is elaborated in detail. The software for new applications of Lebedev type integral transforms and related dual integral equations for the numerical solution of problems of mathematical physics is constructed. The algorithm of numerical solution of some mixed boundary value problems for the Helmholtz equation in wedge domains is developed. Observed examples admitting complete analytical solution demonstrate the efficiency of this approach for applied problems.