Abstract: The paper presents a new method to obtain an approximate solution to plate vibrations problems. The problem is described by a partial differential equation of fourth order. The key idea of the presented approach is to find polynomials (solving functions) that satisfy the considered differential equation identically. In this sense, it is a variant of the Trefftz method. The method is addressed to differential equations in a finite domain. The approach proposed here has some advantages. The first is that the approximate solution (a linear combination of the solving functions) satisfies the equation identically. Secondly, the method is flexible in terms of given boundary and initial conditions (discrete, missing). Thirdly, the solving functions can be used as base functions for several variants of the Finite Element Method. In this case, the approximation is good even for relatively large elements. It means that the approach is suitable for inverse problems. The formulas for solving functions and their derivatives for the plate vibration equation are obtained. The convergence of the method is proved and numerical examples are included.
B I B L I O G R A F I A1. Al-Khatib M.J., Grysa K., Maciąg A., 2008, The method of solving polynomials in the beam vibration problem, Journal of Theoretical and Applied Mechanics, 46, 2, 347-366
2. Ciałkowski M.J., 1999, Solution of inverse heat conduction problem with use new type of finite element base functions, in: Proceedings of the International Symposium on Trends in Continuum Physics, B.T. Maruszewski, W. Muschik and A. Radowicz (Eds.), World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 64-78
3. Ciałkowski M.J., 2003, Thermal and related functions used in solving certain problems of mechanics. Part I - Solution of certain differential equations by means of inverse operations, Studia i materiały. Technika. Uniwersytet Zielonogórski, 3, 7-70 [in Polish]
4. Ciałkowski M.J., Frąckowiak A., 2000, Heat Functions and Their Application for Solving Heat Transfer and Mechanical Problems, Poznan University of Technology Publishers [in Polish]
5. Ciałkowski M.J., Frąckowiak A., 2003, Thermal and related functions used in solving certain problems of mechanics. Part I - Effective determination of inverse operations applied to harmonic functions, Studia i materiały. Technika. Uniwersytet Zielonogórski, 3, 71-98 [in Polish]
6. Ciałkowski M.J., Frąckowiak A., 2004, Application of symbolic operations to the determination of Trefftz functions for a heat flow wave equation, Zeszyty Naukowe Politechniki Poznańskiej, Maszyny Robocze i Transport, 57 [in Polish]
7. Ciałkowski M.J., Frąckowiak A., Grysa K., 2007, Solution of a stationary inverse heat conduction problems by means of Trefftz non-continuous method, International Journal of Heat Mass Transfer, 50, 2170-2181
8. Ciałkowski M.J., Futakiewicz S., Hożejowski L., 1999a, Heat polynomials applied to direct and inverse heat conduction problems, in: Proceedings of the International Symposium on Trends in Continuum Physics, B.T. Maruszewski, W. Muschik and A. Radowicz (Eds.), World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 79-88
9. Ciałkowski M.J., Futakiewicz S., Hożejowski L., 199b, Method of heat polynomials in solving the inverse heat conduction problems, ZAMM, 79, 709-710
10. Ciałkowski M.J. Jarosławski M., 2003, Application of symbolic calculations in generating the solution of the wave equation, Zeszyty Naukowe Politechniki Poznańskiej, Maszyny Robocze i Transport, 56 [in Polish]
11. Futakiewicz S., 1999, Heat Functions Method for Solving Direct and Inverse Heat Conductions Problems, Ph.D. Thesis, Poznan University of Technology [in Polish]
12. Futakiewicz S., Grysa K., Hożejowski L., 1999, On a problem of boundary temperature identification in a cylindrical layer, in: Proceedings of the International Symposium on Trends in Continuum Physics, B.T. Maruszewski, W. Muschik and A. Radowicz (Eds.), World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 119-125
13. Futakiewicz S., Hożejowski L., 1998a, Heat polynomials in solving the direct and inverse heat conduction problems in a cylindrical system of coordinates, in: Advanced Computational Method in Heat Transfer V, A.J. Nowak, C.A. Brebbia, R. Bialecki and M. Zerroukat (Eds.), Computational Mechanics Publications, Southampton, Boston, 71-80
14. Futakiewicz S., Hożejowski L., 1998b, Heat polynomials method in the n-dimensional direct and inverse heat conduction problems, in: Advanced Computational Method in Heat Transfer V, A.J. Nowak, C.A. Brebbia, R. Bialecki and M. Zerroukat (Eds.), Computational Mechanics Publications, Southampton, Boston, 103-112
15. Herrera I., Sabina F., 1978, Connectivity as an alternative to boundary integral equations: Construction of bases, Appl. Math. Phys. Sc., 75, 5, 2059-2063
16. Hożejowski L., 1999, Heat Polynomials and their Application for Solving Direct and Inverse Heat Condutions Problems, Ph.D. Thesis, Kielce University of Technology [in Polish]
17. Jirousek J., 1978, Basis for development of large finite elements locally satisfying all fields equations, Comp. Meth. Appl. Mech. Eng., 14, 65-92
18. Jirousek J., Zieliński A.P., Rabemantantsoa H., Venkatesh A., 1997, Survey of Trefftz-type element formulations, Computers and Structures, 63, 2, 225-242
19. Kołodziej J.A., Zieliński A.P., 2009, Boundary Collocation Techniques and their Application in Engineering, WIT Press, Southampton, Boston
20. Kupradze V.D., 1989, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Publ. Cmp, Amsterdam
21. Maciąg A., 2004, Solution of the three-dimensional wave equation by using wave polynomials, PAMM - Proc. Math. Mech., 4, 706-707
22. Maciąg A., 2005, Solution of the three-dimensional wave polynomials, Mathematical Problems in Engineering, 5, 583-598
23. Maciąg A., 2007a, Two-dimensional wave polynomials as base functions for continuity and discontinuity Finite Elements Method, in: Współczesne technologie i urządzenia energetyczne, J. Taler (Ed.), Krakow, 371-381 [in Polish]
24. Maciąg A., 2007b, Wave polynomials in elasticity problems, Engineering Transactions, 55, 2, 129-153
25. Maciąg A., Wauer J., 2005a, Solution of the two-dimensional wave equation by using wave polynomials, Journal of Engineering Mathematics, 51, 4, 339-350
26. Maciąg A., Wauer J., 2005b, Wave polynomials for solving different types of two-dimensional wave equations, Computer Assisted Mechanics and Engineering Sciences, 12, 87-102
27. Rosenbloom P.C., Widder D.V., Expansion in terms of heat polynomials and associated functions, Trans. Am. Math. Soc., 92, 220-266
28. Trefftz E., 1926, Ein Gegenst¨uk zum Ritz'schen Verfahren, Proceedings 2nd International Congres of Applied Mechanics, Zurich, 131-137
29. Yano H., Fukutani S., Kieda A., 1983, A boundary residual method with heat polynomials for solving unsteady heat conduction problems, Franklin Inst., 316, 291-298
30. Zieliński A.P., Zienkiewicz O.C., 1985, Generalized finite element analysis with T-complete boundary solution functions, Int. J. Numer. Meth. Eng., 21, 509-528 
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