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It discusses the implementation of such a numerical solution for PDE\u2019s presenting the construction of the problem\u2019s algebraic representation and exploring solution mechanisms with different orthogonal polynomial bases. It highlights contexts of opportunity and the advantages of exploring low-rank approximations and well-conditioned linear systems, despite the fact that spectral methods usually give rise to dense and ill-conditioned matrices. It presents , a  numerical library for the solution of integro-differential problems. It shows numerical experiments illustrating the implementations\u2019 accuracy and computational costs. 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