On the solutions of the polynomic Laplacian equation
Author:
George Dassios
Journal:
Quart. Appl. Math. 74 (2016), 643-646
MSC (2010):
Primary 35J05, 31C10, 31C99
DOI:
https://doi.org/10.1090/qam/1451
Published electronically:
July 18, 2016
MathSciNet review:
3539026
Full-text PDF Free Access
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Abstract: In this short communication we demonstrate a representation of the solutions of a partial differential equation, which is a polynomial in the Laplacian, in terms of harmonic functions alone. The idea is based on the Vekua Trasformation, which connects the kernel of the Laplace operator with the kernel of the Helmholtz operator. The representation can be applied to some well-known equations, such as the Brinkman equation in Viscous Hydrodynamics or the equation of Shells in Elasticity.
References
- Antonios Charalambopoulos and George Dassios, On the Vekua pair in spheroidal geometry and its role in solving boundary value problems, Appl. Anal. 81 (2002), no. 1, 85–113. MR 1926803, DOI 10.1080/0003681021000021088
- D. Palaniappan, On some general solutions of transient Stokes and Brinkman equations, Journal of Theoretical and Applied Mechanics 52 (2014), 405–415.
- I. N. Vekua, New methods for solving elliptic equations, North-Holland Series in Applied Mathematics and Mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Translated from the Russian by D. E. Brown; Translation edited by A. B. Tayler. MR 0212370
References
- Antonios Charalambopoulos and George Dassios, On the Vekua pair in spheroidal geometry and its role in solving boundary value problems, Appl. Anal. 81 (2002), no. 1, 85–113. MR 1926803, DOI 10.1080/0003681021000021088
- D. Palaniappan, On some general solutions of transient Stokes and Brinkman equations, Journal of Theoretical and Applied Mechanics 52 (2014), 405–415.
- I. N. Vekua, New methods for solving elliptic equations, translated from the Russian by D. E. Brown, translation edited by A. B. Tayler, North-Holland Series in Applied Mathematics and Mechanics, Vol. 1, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0212370
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Additional Information
George Dassios
Affiliation:
Division of Applied Mathematics, Department of Chemical Engineering, University of Patras – and – ICE/TH, FORTH, Patras, Greece
MR Author ID:
54715
Email:
gdassios@chemeng.upatras.gr
Received by editor(s):
November 20, 2015
Published electronically:
July 18, 2016
Article copyright:
© Copyright 2016
Brown University
