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The paper demonstrates a specific power series expansion technique used to obtain the approximate solution of the two-dimensional wave equation in some unusual cases. The solution for inhomogeneous wave equation, for more complicated shape geometry of the body, discrete boundary conditions and a membrane whose thickness is not constant is shown. As solving functions (Trefftz functions), so-called wave polynomials are used. Recurrent formulas for the particular solution are obtained. Some examples are included.
Rocznik
Tom
Strony
364--378
Opis fizyczny
Bibliogr. 18 poz., il., wykr.
Bibliografia
- [1] E. Trefftz. Ein Gegenstük zum Ritzschen Verfahren. In: Proceedings 2nd International Congres of Applied Mechanics, 131-137, Zurich 1926.
- [2] A.P. Zieliński and I. Herrera. Trefftz method: fitting boundary conditions. Int. J. Num. Meth. Eng., 24: 871-891, 1987.
- [3] P.C. Rosenbloom and D.V. Widder. Expansion in terms of heat polynomials and associated functions. Trans. Am. Math. Soc., 92: 220-266, 1956.
- [4] H. Yano, S. Fukutani and A. Kieda. A boundary residual method with heat polynomials for solving unsteady heat conduction problems. Journal of the Franklin Institute, 316 (4): 291-298, 1983.
- [5] S. Futakiewicz and L. Hożejowski. Heat polynomials method in the n-dimensional direct and inverse heat conduction problems. In: A.J. Nowak, CA. Brebbia, R. Bielecki and M. Zerroukat, eds., Advanced Computational Method in Heat Transfer V, 103-112, Southampton UK and Boston USA: Computational Mechanics Publications 1998.
- [6] L. Hożejowski. Heat polynomials and their application for solving direct and inverse heat condutions problems (PhD-Thesis), (in Polish), 115 Kielce: University of Technology, 1999.
- [7] S. Futakiewicz and L. Hożejowski. Heat polynomials in solving the direct and inverse heat conduction problems in a cylindrical system of coordinates. In: A.J. Nowak, CA. Brebbia, R. Bielecki and M. Zerroukat. eds., Advanced Computational Method in Heat Transfer V, 71-80 and Boston USA Computational Mechanics Publications, Southampton UK, 1998.
- [8] S. Flitakiewicz, K. Grysa and L. Hożejowski. On a problem of boundary temperaturę Identification in a cylindrical layer. In: B.T. Maruszewski, W. Muschik and A. Radowicz eds., Proceedings of the International Symposium on Trend’s in Continuum Physics, 119-125, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong , 1999.
- [9] S. Futakiewicz. Heat functions method for solving direct and inverse heat condutions problems (PhD-Thesis) (in Polish), 120, University of Technology, Poznań 1999.
- [10] P. Johansen, M. Nielsen and O.F. Olsen. Branch Points in One-Dimensional Gaussian Scale Space. Journal of Mathematical Imaging and Vision, 13: 193-203 2000.
- [11] M.J. Ciałkowski, S. Futakiewicz and L. Hożejowski. Method of heat polynomials in solving the inverse heat conduction problems. ZAMM 79: 709-710, 1999.
- [12] M.J. Ciałkowski, S. Futakiewicz and L. Hożejowski. Heat polynomials applied to direct and inverse heat conduction problems. In: B.T. Maruszewski, W. Muschik and A. Radowicz eds., Proceedings of the International Symposium on Trends in Continuum Physics, 79-88, World Scientific Publishing Singapore, New Jersey, London, Hong Kong, 1999.
- [13] M.J. Ciałkowski. Solution of inverse heat conduction problem with use new type of finite element base functions. In: B.T. Maruszewski, W. Muschik and A. Radowicz eds., Proceedings of the International Symposium on Trends in Continuum Physics, 64-78, World Scientific Publishing Singapore, New Jersey, London, Hong Kong 1999.
- [14] M.J. Ciałkowski and A. Frąckowiak. Heat functions and their application for solving heat transfer and mechanical problems. (in Polish), 360 University of Technology Publishers Poznań 2000.
- [15] I.N. Sneddon. Elements of Partial Differential Equations, 423. PWN, Warsaw 1962.
- [16] A. Maciąg and J. Wauer. Solution of the two-dimensional wave equation by using wave polynomials, Journal of Engrg. Math. 51: (4) 339-350, 2005.
- [17] E.B. Magrab. Vibrations of elastic structural members, 320. Maryland USA: Sijthoff & Noordhoff, 1979.
- [18] E. Kącki. Partial differential equations in phisics and engineering problems (in Polish), 506. WNT, Warsaw 1989
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Bibliografia
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bwmeta1.element.baztech-article-BPB2-0016-0004