Fractional Chebyshev spectral scheme for time-fractional seventh-order KdV equations with non-smooth solutions

Source abstract

One of the key challenges in solving time-fractional evolution equations, such as the seventh-order KdV equations, is the non-smooth behavior of the solution in the time direction, which arises from the weakly singular kernel of the fractional derivative. This non-smoothness often limits the performance of standard polynomial-based methods in accurately capturing the solution near the initial time. To address this, we adopt a fully spectral collocation method in both space and time, which offers high-order accuracy. In particular, we employ fractional-order Chebyshev functions as temporal basis functions, which are well-suited for approximating the singular behavior of fractional-order solutions and provide improved resolution in the early stages of the solution profile. Shifted Chebyshev polynomials are used in the spatial direction to maintain spectral accuracy. Our method transforms the original problem into a system of algebraic equations using an operational matrix approach, and the resulting numerical scheme is validated through comparisons with other existing numerical methods, demonstrating the accuracy and the superiority of the proposed spectral framework.