Summary
This article reports a method to handle integrated forms of linear ordinary differential equations by means of matrix operator expressions, which apply to integral terms and non-constant coefficients. The method avoids the use of a numerical grid and includes the treatment of boundary conditions and inhomogeneous terms. It can be regarded as a mechanized version of the τ-method. The application of the method to equations in integrated form leads to linear algebraic systems with better condition compared to the differential operator. Therefore, the method permits the application of iterative methods in order to solve the linear systems. Its effective application is demonstrated by two examples. Furthermore, the method is extended to linear parabolic problems. Finally, a solution of the Orr-Sommerfeld equation is presented to indicate the treatment of eigenvalue problems.
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Hiegemann, M. Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations. Acta Mechanica 122, 231–242 (1997). https://doi.org/10.1007/BF01182002
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DOI: https://doi.org/10.1007/BF01182002