Download Haar Wavelets: With Applications by Ülo Lepik, Helle Hein PDF

By Ülo Lepik, Helle Hein

This is the 1st e-book to give a scientific assessment of functions of the Haar wavelet technique for fixing Calculus and Structural Mechanics difficulties. Haar wavelet-based ideas for a variety of difficulties, similar to a number of differential and fundamental equations, fractional equations, optimum keep an eye on conception, buckling, bending and vibrations of elastic beams are thought of. Numerical examples demonstrating the potency and accuracy of the Haar technique are supplied for all solutions.

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Now let us put together a Haar wavelet solution with the variable stepsize. We shall assume that the lengths of the subintervals fulfill the condition Δxl = qΔxl−1 , where q < 1 and l = 1, . . , 2M. By summing up all the lengths of the subintervals we obtain Δx1 (1 + q + q 2 + · · · + q 2M−1 ) = 1. 46) Since the sum of this geometric series is 1 + q + q 2 + · · · + q 2M−1 = we obtain Δx1 = 1 − q 2M 1−q 1−q . 48) The grid points are x˜l = 1 − ql , l = 0, 1, . . , 2M. 9). 41). We have to satisfy the boundary condition y(1) = 0.

001 the solution for q = 1 remains unstable in spite of the increasing number of collocation points. 5 Nonlinear Equations The n-th order nonlinear ODE has the form F(x, y, y ≥ , y ≥≥ , . . 54) 34 3 Solution of Ordinary Differential Equations (ODEs) where F is a nonlinear function. 55) i=1 and integrate this equation n-times. 54). By doing this we get a system of nonlinear equations φl (a1 , a2 , . . , a2M ) = 0, l = 1, 2, . . 56) from which the wavelet coefficients ai can be calculated. 56) the Newton method is applied.

2 /H, c˜ = −b. For correcting these values two Newton steps were needed. 626e − 5. 3]. In this phase the functions y1 , y3 decrease very slowly, therefore we take y˜1 = y1 (δ)E, y˜2 = y2 (δ)E. 20) we find y˜3 = y3 (δ)E + 3 · 107 y2 (δ)2 x. 138e − 4. 129e − 4. These data are in good accordance with our results. It is interesting to note that in the second phase, only the first of the wavelet coefficients is significant. 0006, . . 0092, . . 0006, . . All other coefficients are considerably smaller.

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