A Spectral Tau Algorithm Using Shifted Convolved Fibonacci Polynomials for Singular Functional Models

In this paper, for some second-order singular functional differential equations (SOSFDEs), a tau-type spectral technique is built using non-origin-symmetric polynomials called shifted convolved Fibonacci polynomials as the approximation basis. Derivative formulae, integral identities required for the tau projection, power-form and inverse representations, and several operational relations are derived for this basis. The singular term and the shifted arguments can be handled directly within an algebraic framework thanks to these formulas, eliminating the need to introduce a grid on the computing interval. For the unknown coefficients, the resultant approximation yields a finite system. The uniform convergence of the truncated shifted convolved Fibonacci expansion is established, together with error estimates and a stability result for the proposed approximation. To demonstrate the procedure’s accuracy, three cases are utilized as benchmarks. As the truncation order increases, the reported errors gradually diminish, and the computed solutions correspond closely with the exact answers. Further evidence of the reliability and efficiency of the suggested shifted convolved Fibonacci tau procedure is provided by comparisons with existing numerical data.