A Shifted Third‐Kind Chebyshev Spectral Method for Nonlinear Variable‐Order Fractional Differential Equations

This paper presents a spectral collocation method based on shifted Chebyshev polynomials of the third kind (SCP3) for the numerical solution of variable‐order fractional differential equations (VO‐FDEs). The SCP3 basis is employed to construct accurate approximations of the unknown solution over the interval [0, 1], exploiting its orthogonality and favorable approximation properties. An operational matrix formulation is developed to represent both integer‐order derivatives and VO Caputo fractional derivatives in a unified algebraic framework. This formulation transforms the original VO‐FDEs into a system of algebraic equations, enabling efficient numerical implementation. A theoretical analysis is provided, including error representation and convergence analysis. The proposed method exhibits algebraic convergence for sufficiently smooth solutions and spectral convergence for analytic functions. The applicability and performance of the method are demonstrated through several numerical examples, including linear, nonlinear, and coupled variable‐order fractional systems.