An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations. (English) Zbl 1162.65040

Summary: A novel single-term Haar wavelet series (STHWS) method is implemented for the solution of the Duffing equation and Painlevé’s transcendents (PI and PII). The results, in the form of a block pulse and a discrete solution, are presented. Unlike classical numerical schemes, the STHWS method has no restrictions on the coefficients of the Duffing equation as regards its solution. PI and PII are analysed as regards their solutions, up to nearest singularities (poles), using the STHWS. Also, an efficient computational implementation shows the remarkable features of wavelet based techniques.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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[1] Bender, Carl M.; Orszag, Steven A., Advanced mathematical methods for scientists and engineers, (1987), McGraw-Hill International · Zbl 0417.34001
[2] Byrne, G.; Hall, C., Numerical solution of systems of non-linear algebraic equations, vol. 2, (1973), Academic Press New York
[3] Ince, E.L., Ordinary differential equations, (1956), Dover New York · Zbl 0063.02971
[4] Sachdev, P.L., Nonlinear ordinary differential equation and their application, (1991), Marcel Dekker · Zbl 0722.34001
[5] Wojtaszezyk, P., A mathematical introduction to wavelets, (1997), Cambridge University Press
[6] Balachandran, K.; Muragesan, K., Analysis of different systems via single-term Walsh series method, Int. J. comput. math., 33, 171-179, (1989) · Zbl 0752.65056
[7] Carrol, J., A matricial exponentially fitted scheme for the numerical solution of stiff initial value problems, Comput. math. appl., 26, 57-64, (1993) · Zbl 0789.65055
[8] Chen, C.F.; Hsiao, C.H., Haar wavelet method for solving lumped and distributed parameter systems, IEE proc. control theory appl., 144, 1, 87-94, (1997) · Zbl 0880.93014
[9] Fair, W.G.; Luke, Y.L., Rational approximations to the generalized Duffing equation, Internat. J. non-linear mech., 1, 209-216, (1966) · Zbl 0171.36303
[10] Hsiao, C.H., Haar wavelet approach to linear stiff systems, Math. comput. simul., 64, 561-567, (2004) · Zbl 1039.65058
[11] Ludeke, C.A.; Wagner, W.S., The generalized Duffing equation with large damping, Internat. J. non-linear mech., 3, 383-395, (1968) · Zbl 0162.12402
[12] Pavlov, B.V.; Rodionova, O.E., Numerical solution of systems of linear ordinary differential equations with constant coefficients, Comput. math. phys., 34, 535-539, (1994) · Zbl 0820.65037
[13] Rao, G.P.; Palanisamy, K.R.; Srinivasan, T., Extension of computation beyond the limit of initial normal interval in Walsh series analysis of dynamical systems, IEEE trans. automat. control, 25, 317-319, (1980) · Zbl 0442.93021
[14] Sannuti, P., Analysis and synthesis of dynamical systems via block- pulse functions, Proc. IEE, 124, 569-571, (1977)
[15] Sepehrian, B.; Razzaghi, M., Solution of time-varying singular nonlinear systems by single-term Walsh series, Math. problems eng., 3, 129-136, (2003) · Zbl 1175.93100
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