The Use of Spectral Methods in Solving Boundary Value Problems and the Study of Their Numerical Efficiency

Sadiq Aziz Hussein

doi:10.71229/n5ea2s70

Authors

Sadiq Aziz Hussein Third Rusafa Education Directorate

DOI:

https://doi.org/10.71229/n5ea2s70

Keywords:

Boundary value problems, , spectral methods,, Fourier series,, Chebyshev polynomials,, error analysis,, convergence rate,, numerical stability,, numerical efficiency.

Abstract

This research aims to investigate the use of spectral methods in solving boundary value problems and to analyze their numerical efficiency in terms of accuracy, convergence rate, and numerical stability, with a comparison to some traditional numerical methods. Boundary value problems are among the fundamental topics in applied mathematics due to their central role in modeling many physical and engineering phenomena. However, the complexity of differential equations and the coupling of boundary conditions often make it difficult to obtain exact analytical solutions, which necessitates the use of numerical approaches. The study adopts an analytical approach to present the theoretical framework of boundary value problems, clarifying their concept and the types of associated boundary conditions, in addition to reviewing traditional numerical methods and the limitations of their efficiency. A numerical–applied approach is also employed to examine spectral methods, particularly those based on Fourier series and Chebyshev polynomials, by explaining their mathematical foundations and the mechanisms for implementing different boundary conditions. The applied part includes solving selected models of boundary value problems using spectral methods, conducting a detailed analysis of numerical error and convergence rate, studying numerical stability, and evaluating numerical efficiency through comparisons of accuracy and computational time with some conventional methods. The results demonstrate that spectral methods achieve exponential convergence when dealing with smooth solutions, enabling high accuracy to be obtained using a relatively small number of degrees of freedom, which positively affects computational cost. The study also shows that Fourier-based spectral methods are more suitable for problems with periodic boundary conditions, whereas Chebyshev-based methods exhibit higher efficiency in handling non-periodic problems. The research concludes that spectral methods represent an effective numerical option for solving high-accuracy boundary value problems, emphasizing the importance of proper mathematical formulation and accurate implementation of boundary conditions to ensure numerical stability and efficiency.

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