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Mellin transform; expansion of Laguerre polynomials; numerical inversion; discrete least squares approximation; numerical examples
In order to use the well known representation of the Mellin transform as a combination of two Laplace transforms, the inverse function $g(r)$ is represented as an expansion of Laguerre polynomials with respect to the variable $t=ln\ r$. The Mellin transform of the series can be written as a Laurent series. Consequently, the coefficients of the numerical inversion procedure can be estimated. The discrete least squares approximation gives another determination of the coefficients of the series expansion. The last technique is applied to numerical examples.
[1] I. N. Sneddon: Fourier Transforms. McGraw-Hill, New York, 1951. MR 0041963
[2] D. Bogy: On the Problem of Edge-Bonded Elastic Quarter-Planes Loaded at the Boundary. Int. Journ. Sol. Structures, 6 (1970), 1287-1313. DOI 10.1016/0020-7683(70)90104-6 | Zbl 0202.25001
[3] G. Tsamasphyros, P. S. Theocaris: Numerical Inversion of Mellin Transforms. BIT, 16 (1976), 313-321. DOI 10.1007/BF01932274 | MR 0423763 | Zbl 0336.65059
[4] V. I. Krylov, N. S. Skoblya: Handbook of Numerical Inversion of Laplace Transform. Minsk (1968) and Israel program for scientific translations, Jerusalem, 1969. MR 0391481
[5] F. Tricomi: Transformazione di Laplace e polinomi de Laguerre. R. C. Accad. Naz. Lincei, Cl. Sci. Fis. 1a, 13 (1935), 232-239 and 420-426.
[6] G. Doetch: Handbuch der Laplace Transformation. Verlag Birkhäuser, Basel, 1950.
[7] A. Papoulis: A New Method of Inversion of the Laplace Transform. Quart. Appl. Math. 14 (1956), 405-414. DOI 10.1090/qam/82734 | MR 0082734
[8] W. T. Weeks: Numerical Inversion of Laplace Transforms Using Laguerre Functions. Journal ACM. 13 (1966), 419-426. DOI 10.1145/321341.321351 | MR 0195241 | Zbl 0141.33401
[9] R. Piessens, M. Branders: Numerical Inversion of the Laplace Transform Using Generalised Laguerre Polynomials. Proc. IEE 118 (1971), 1517-1522. MR 0323084
[10] R. Piessens: A Bibliography on Numerical Inversion of the Laplace Transform and Applications. Jour. Comput. Appl. Mathern. 1 (1975), 115-128, DOI 10.1016/0771-050X(75)90029-7 | MR 0375743 | Zbl 0302.65092
[11] R. Piessens, F. Poleunis: A Numerical Method for the Integration of Oscillatory Functions. BIT, 11 (1971), 317-327. DOI 10.1007/BF01931813 | MR 0288959 | Zbl 0234.65026
[12] A. Alaylioglu G. Evans, J. Hyslop: Automatic Generation of Quadrature Formulae for Oscillatory Integrals. Соmр. Jour. 18 (1975), 173-176 and 19 (1976), 258-267. MR 0375747
[13] T. Vogel: Les fonctions orthogonales dans les problèmes aux limites de la physique Mathematique. CNRS, 1953. MR 0060053 | Zbl 0052.29003
[14] A. Erdelyi W. Magnus F. Oberthettinger F. G. Tricomi: Tables of Integral Transforms. McGraw-Hill, New York, 1954.
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