Computational and numerical analysis of differential equations using spectral based collocation method.
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In this thesis, we develop accurate and computationally eﬃcient spectral collocation-based methods, both modiﬁed and new, and apply them to solve diﬀerential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of diﬀerential equations deﬁned over simple geometries. Procedurally, these methods entail transforming the gov erning diﬀerential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deﬁciency, the present study explores the development of numerical algorithms for solving ordinary and partial diﬀerential equations deﬁned over simple geometries. The solutions of the diﬀerential equations considered are approximated using interpolating polynomials that satisfy the given diﬀerential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve diﬀerential equations deﬁned on large computational domains much eﬀort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con ﬁrms the superiority of these multiple domain techniques in terms of accuracy and computational eﬃciency over the single domain approach when applied to problems deﬁned over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across diﬀerent dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary diﬀerential equations. Subsequent chapters then build on solutions to partial diﬀerential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the diﬀerent numerical algorithms. The numerical results of the study conﬁrm that incorporating domain decomposition techniques in spectral collocation methods work eﬀectively for all dimensions, as we report highly accurate results obtained in a computationally eﬃcient manner for problems deﬁned on large do mains. The ﬁndings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter.