Browsing by Author "Wlie, Saeida"
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Item PSEUDOSPECTRAL METHODS WITH VARIOUS BASIS FUNCTIONS AND APPLICATIONS TO QUANTUM MECHANICS(2022-02-28) Wlie, Saeida; Erhan, İnciIn this thesis, we studied the pseudospectral methods and their application to the so lution of eigenvalue problems associated with ordinary differential equations. In par ticular, we considered second order differential equations and a specific example, the Schrodinger equation for quantum dynamical systems with polynomial potentials. ¨ After an introduction to self adjoint eigenvalue problems and the Schrodinger equation ¨ for particles, in the presence of polynomial potentials, we recollected some impor tant properties of Lagrange interpolation and orthogonal polynomials. We presented a method to compute the zeros of an orthogonal polynomial of arbitrary degree by means of a symmetric tridiagonal matrix eigenvalue problem. We constructed the particular symmetric tridiagonal matrices for computation of the zeros of Hermite, Associated Laguerre, Chebyshev and Legendre polynomials. After that, we explained in details the pseudospectral schemes using Hermite and Associated Laguerre polynomials by studying some published articles. We also made substitutions on the independent variable in order to transform infinite interval to a finite one and derived pseudospectral formulations using Chebyshev and Legendre polynomials. As a specific example, we applied the pseudospectral methods using the four types of orthogonal polynomials mentioned above to the Schrodinger equation for quantum ¨ dynamical systems with polynomial potentials. We compared our numerical results with the numerical results obtained previously by other authors and made comments about the efficiency of our method.