Time-fractional nonlinear evolution of dynamic wave propagation using the Burgers’ equation

Fractional derivatives are crucial in diverse contexts, offering a means to extend classical derivatives to noninteger orders. This expansion of calculus enables a more detailed understanding of complex behaviors in scientific, engineering, and mathematical disciplines. In this study, we use theoretical and numerical analyses to thoroughly examine the time-fractional Burgers’ equation. Our main emphasis is on deriving time-decay estimates for solutions within a bounded domain. To determine optimal time-decay rates, we introduce a linear compact difference scheme by integrating an \(L_1\) $L_1$ discretization formula for the Caputo derivative and compact difference operators with spatial derivatives. We provide a detailed analysis encompassing existence and uniqueness, and an error estimate of solutions under the \(\Vert \cdot \Vert _{\infty }\) ${\Vert ·\Vert }_{\infty }$ norm for the proposed scheme. Comprehensive numerical experiments highlight the efficiency and robustness of our approach, ensuring reliability for long-time simulations. Furthermore, numerical results are scrutinized to pinpoint the optimal time-decay rate. This work explores the asymptotic behavior of solutions to the time-fractional Burgers’ equation and the development of linear high-order accuracy difference methods for nonlinear time-fractional equations.