Spectral Lanczos' tau method for systems of nonlinear integro-differential equations
For researchers in numerical analysis and applied mathematics, this provides a more accurate and stable method for solving nonlinear integro-differential systems, though it is an incremental extension of existing techniques.
The paper extends the spectral Lanczos' tau method to systems of nonlinear integro-differential equations, introducing improvements for numerical stability and accuracy. The method significantly outperforms other numerical approximations and tau implementations on a set of test problems.
In this paper an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis. All these improvements ensure numerical stability and accuracy in the approximate solution. Exposed in detail, this novel approach is able to significantly outperform numerical approximations with other methods as well as different tau implementations. Numerical results on a set of problems illustrate the impact of the mathematical techniques introduced.