Generalized Lucas Polynomial Method for Solving Two-classes of ψ-fractional Differential Equations

چکیده مقاله

ψ-fractional differential equations (ψ-FDEs) extend the capabilities of standard fractional differential equations by introducing a general function ψ in the fractional derivative operator. We can recover various known fractional derivatives such as Riemann-Liouville, Caputo, Hadamard, and others by selecting specific forms of ψ. This provides a flexible framework for modeling different types of non-local behaviors. Hence, this study is devoted to designing a new computational scheme to solve ψ-FDEs and ψ-fractional integro-differential equations (ψ-FIDEs) with ψ-Caputo derivative. We use the generalized Lucas polynomials (G-LPs) and the collocation method to develop the desired technique. To do this, we propose a ψ-Caputo derivative operator for G-LPs. Subsequently, by employing the collocation method, and the mentioned required preliminaries, the addressed problems are transformed into systems of algebraic equations, which can be solved through Newton’s iterative method. Numerical results and comparative analyses illustrate that the proposed method exhibits a high accuracy and efficiency. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2026.