A new Weber-type transform
Authors:
R. K. Michael Thambynayagam and Tarek M. Habashy
Journal:
Quart. Appl. Math. 61 (2003), 485-493
MSC:
Primary 44A15; Secondary 86-08
DOI:
https://doi.org/10.1090/qam/1999833
MathSciNet review:
MR1999833
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Abstract: In this paper we introduce a new Weber-type transform pair for the representation of a function $f\left ( r \right )$ defined over the domain $a \le r < \infty$ and which satisfies the Robin mixed boundary condition $f\left ( a \right ) + \lambda f’\left ( a \right ) = 0$. The orthogonality relationships of the transform kernels are derived in both the spatial and the spectral domains as well as Parseval’s theorem. We apply this new Weber-type transform pair to solve a mixed boundary value problem in a system of planar layers.
- H. Weber, Ueber eine Darstellung willkürlicher Functionen durch Bessel’sche Functionen, Math. Ann. 6 (1873), no. 2, 146–161 (German). MR 1509813, DOI https://doi.org/10.1007/BF01443190
G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1980)
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
H. Hankel, Die Fourier’schen Reihen und Integrale für Cylinder functionen (original memoirs 1869), Math. Ann. VIII, 467 (1875)
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)
F. Kuchuk and T. Habashy, Pressure behavior of laterally composite reservoirs, SPE Journal on Formation Evaluation 12, 47 (1997)
H. Weber, Ueber eine Darstellung willkürliccher Functionen durch Bessel’sche Functionen, Math. Ann. VI, 146 (1873)
G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1980)
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations (Oxford University Press, Oxford, 1962)
H. Hankel, Die Fourier’schen Reihen und Integrale für Cylinder functionen (original memoirs 1869), Math. Ann. VIII, 467 (1875)
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)
F. Kuchuk and T. Habashy, Pressure behavior of laterally composite reservoirs, SPE Journal on Formation Evaluation 12, 47 (1997)
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Article copyright:
© Copyright 2003
American Mathematical Society
