By Ülo Lepik, Helle Hein
This is the 1st publication to offer a scientific evaluate of functions of the Haar wavelet strategy for fixing Calculus and Structural Mechanics difficulties. Haar wavelet-based suggestions for a variety of difficulties, corresponding to a number of differential and critical equations, fractional equations, optimum keep an eye on idea, 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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Extra resources for Haar Wavelets: With Applications
3. Small oscillations of the curve y2 indicate instability of the solution. Let us pass to the Haar wavelet method. 65e − 5; after that the function y2 decreases very slowly. 20) separately for the regions x √ [0, δ] and x √ [δ, xmax ]. The wavelet solution is sought in the form y1∈ = aH, y1 = y1 (0)E + aP1 , y2∈ = bH, y2 = y2 (0)E + bP1 , y3∈ = cH, y3 = y3 (0)E + cP1 . 2 . 21), these vectors are functions of the wavelet coefficients a, b, c. We have to find such values a, b, c for which F1 = F2 = F3 = 0.
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. 42) we have y0≥ = −a P2 |x=1 . 50) Let us introduce the row vector K with the components Ki = 1 [1 − ξ1 (i)]2 − 2[1 − ξ2 (i)]2 + [1 − ξ3 (i)]2 . 4 Error estimates for Eq. 5 Error estimates for Eq. 6) we find y0≥ = −a K T , where the index T denotes transposition. 52) where the symbol ⊗ denotes the Kronecker tensor product. 53) from which the wavelet coefficients are calculated.
84) Here F and G denote the coefficient vectors. 81) and satisfying this system in the collocation points we get a linear system for calculating F and G. This approach has been followed in many papers. First we discuss here the papers by Hsiao and his coworkers in which time varying systems were investigated [13–15]. 85) where E(t) is a singular matrix E(t) ∞ R n×n , the state variable is x(t) ∞ R n , control variable u(t) ∞ R q , f ∞ R n denotes a nonlinear function. The response x(t), t ∞ (0, 1) is required to be found.
Haar Wavelets: With Applications by Ülo Lepik, Helle Hein