On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method

Abstract : The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C 0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler-Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.
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Journal of Applied Mechanics, American Society of Mechanical Engineers, 2013, <10.1115/1.4023637>
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Simon Jones, Mathias Legrand. On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method. Journal of Applied Mechanics, American Society of Mechanical Engineers, 2013, <10.1115/1.4023637>. <hal-01330602>

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