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Abstract To avoid the undesired jump or wiggle phenomenon near the boundary points when the wavelet-based method is employed to solve a boundary-value problem, this paper presents a set of modified scaling base functions through the interval extension of an unknown continuous function defined in a finite interval on the basis of the Taylor series expansion associated with the arbitrary boundary conditions. After that, an approximate scheme of the function is proposed by the modified scaling base functions. According to the generalized-Gaussian-quadrature method in wavelet analysis, which was developed by the last two authors of this paper, the expansion constants in the approximation of arbitrary nonlinear term of the unknown function can be explicitly expressed in finite terms of the expansion ones of the approximation of the unknown function. Once the wavelet-Galerkin method on the basis of the approximation is employed to solve the nonlinear differential equation with the nonlinear term(s) of a finite beam structure with arbitrary boundary conditions, it is found that the solution has the closure property and the virtue of easy implement in calculation of solving a strong nonlinear problem. The numerical results obtained in solving two cases of large deflected beams with different nonlinear characterization, i.e., either integer order or non-integer order nonlinear terms or both, indicate that this approach has high accuracy.
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Received: 01 February 2010
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2011, Vol. 32 
