Spectral Methods for different classes of Partial Differential Equations

Abstract

The thesis touches numerical solutions and wave propagation analysis using spectral methods for different classes of partial differential equations (PDEs) having applications in various fields. We address the solution of both linear and nonlinear PDEs. The different problems studied in the thesis are the one dimensional (1-D) wave equation for uniform and varying cross-sectional area, the coupled one dimensional Timoshenko beam with uniform and varying cross-section, the three dimensional linear Heat equation having real solution, the two dimensional linear Schroedinger equation having oscillatory solution, one dimensional nonlinear Korteg-de-Vries equation (KdV), two dimensional nonlinear Schroedinger (NLS) equation, and finally the coupled one-dimensional (1-D), two-dimensional (2-D) and three dimensional (3-D) quasicrystals with four, five and six variables. The major objective of the work is to bring out the versatility of the spectral methods to solution for the above listed equations. The solutions to these partial differential equations discussed in the thesis has been obtained by approximating the unknown function using spectral functions such as Legendre or Chebyshev polynomials as basis function in the frequency domain and also in the time domain. For the nonlinear partial differential equations, we have obtained solutions using Fourier spectral functions along the spatial direction in time domain. Another important area that was studied in the thesis is the wave propagation in the frequency domain. Wave propagation is a transient behaviour resulting from short duration loading, which have high frequency content. The key factor in the wave propagation is the propagating velocity of the waves, the level of attenuation of their response and their wavelengths. It is a multi-modal phenomenon and hence the analysis becomes difficult if the problem is solved in the time domain. W e have discussed the wave propagation analysis in rods, Timoshenko beam and 1-D, 2-D and 3-D quasicrystals. We have also shown that the numerical spectral methods effectively evolve the physical behaviour of the above equations. We also consider wave propagation in different quasicrystals on a beam type structure using 2-D spectral element formulation. An Aluminium Cantilever beam is reinforced with a layer of quasicrystal under different orientation and the wave propagation characteristics of the hybrid structure is studied. The analysis is performed using frequency domain spectral finite element formulation. For all the combinations of quasicrystal Aluminium beam combination, there is substantial suppression of responses both for the axial and the bending responses.