Solving integro-differential problems with the spectral Lanczos' Tau method

Solving integro-differential problems with the spectral Lanczos' Tau method


2022-05-27 - 16:30

Paulo Vasconcelos

Faculdade de Economia da Universidade do Porto

The spectral Tau method, introduced by Cornelius Lanczos, is extremely accurate for solving differential problems and numerical software using this method are often faster than other more standard techniques. It is a spectral method thus ensuring excellent error properties - exponential convergence, whenever the solution is smooth. Originally developed for ordinary differential problems with polynomial coefficients, the method has been casually extended for application in integro-differential, nonlinear, partial differential and fractional integro-differential problems. Nevertheless, in all these extensions the Tau method is tuned for the approximation of specific problems and not offered as a general-purpose numerical tool. To fill this gap we are developing Tau Toolbox, a mathematical software library to produces approximate polynomial solutions of integro-differential equations via the spectral Lanczos' Tau method. This numerical library, in a similar vein to the Chebfun package, enables a symbolic syntax to be applied to objects to manipulate and solve differential problems with ease and accuracy. The library is explained and its application to various problems is illustrated.

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