Abstract
The paper presents a specific technique of solving the non-homogenous wave equation with the use of Trefftz functions for the wave equation. The solution was presented as a sum of a general integral and a particular integral. The general integral was expressed in the form of a linear combination of Trefftz functions for the wave equation. In order to obtain the particular integral polywave functions were used. They were generated by using the inverse operator L-1 of the equation taking into consideration the Trefftz functions.
Keywords:
polywave functions, Trefftz functions, wave polynomials, wave equationReferences
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[2] I. Babuška, T. Strouboulis. The Finite Element Method and its Reliability. Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2001.
[3] M. Ciałkowski, A. Frąckowiak. Heat functions and their application to solving heat conduction and mechanics problems (in Polish). Poznan University of Technology Publishing House, 2000.
[4] M. Ciałkowski, A. Frąckowiak. Heat and related functions in solving selected mechanics problems. Part I: Solution of certain differential equations by means of inverse operations (in Polish). Studia i materiały LIII. Technika 3, pp. 7–70, Uniwersytet Zielonogórski, 2003.
[5] M. Ciałkowski, A. Frąckowiak. Thermal and related functions used in solving certain problems of mechanics. Part II: Effective determination of inverse operations applied to harmonic functions (in Polish). Studia i materiały LIII. Technika 3, pp. 71–98, Uniwersytet Zielonogórski, 2003.
[6] R. Heuer, H. Irschik. A Boundary Element Method for Eigenvalue Problems of Polygonal Membranes and Plates. Acta Mechanica, 66: 9–20, 1987.
[7] A. Maciąg. Three-dimensional wave polynomials. Mathematical Problems in Engineering, 5: 583-598, 2005.
[8] A. Maciąg, J. Wauer. Wave polynomials for solving different types of two-dimensional wave equations.
Computer Assisted Mechanics and Engineering Sciences (CAMES), 12: 363–378, 2005.
[9] A. Maciąg, B. Maciejewska, M. Sokała. 2D wave polynomials as base functions in modified FEM. Computer Assisted Mechanics and Engineering Sciences (CAMES), 15: 265–278, 2008.
[10] E.B. Magrab.Vibrations of elastic structural members, SIJTHOFF & NOORDHOFF, Maryland USA, 1979.
[11] W.J. Mansur, D. Soares Jr., M.A.C. Ferro. Initial conditions in frequency-domain analysis: the FEM applied toscalar wave equation. Journal of Sound and Vibration, 270: 767–780, 2004.
[12] M.A. Rincon, R.D. Rodrigues. Numerical solution for the model of vibrating elastic membrane with moving boundary. Communications in Nonlinear Science and Numerical Simulation, 12(6): 1089–1100, 2007.
[13] M. Sokała. Solution of two-dimensional wave equation using some form of the Trefftz Functions. Computational Methods in Science and Technology, 14(2), 2008.
[14] M. Sokała. Analytical and numerical method of solving heat conduction problems with the use of heat functions and inverse operations. PhD thesis (in Polish), Poznań, 2004.
[15] D.L. Young, M.H. Gu, C.M. Fan. The time-marching method of fundamental solutions for wave equations. Engineering Analysis with Boundary Elements, 33: 1411–1425, 2009.
[16] Z.-C. Li, T.-T. Lu, H.-S. Tsai, A.H.-D. Cheng. The Trefftz method for solving eigenvalue problems. Engineering Analysis with Boundary Elements, 30: 292–308, 2006.
