| Summary: | Abstract In this article, we introduce a new operational matrix of the fractional-order derivative that depends on the Caputo operator of differentiation and first-kind Dickson polynomials. The established matrix, in addition to the Tau spectral method, will be used for solving three important applications in the fractional-order forms: the Bagley-Torvik problem, the Lane-Emden type equation, and Bratu’s problem. The primary approach for addressing these applications involves transforming the fractional-order problem and its conditions into a system of algebraic equations with unknown coefficients. The error estimation and convergence analysis of the proposed method will be thoroughly examined. Our proposed technique will be applied through some examples with comparisons. The introduced numerical results support the theoretical ones, in addition to demonstrating the accuracy and applicability of the suggested method.
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