Efficiently solving fractional delay differential equations of variable order via an adjusted spectral element approach

This paper presents a new approach for solving fractional delay differential equations of variable order using the spectral element method. The proposed method overcomes the limitations of traditional spectral methods, such as poor approximation in long intervals and inefficiency in high degrees. By introducing a variable order differentiation matrix and using basic Lagrangian functions to approximate the solution in each element, the method achieves high accuracy and efficiency. A penalty method is also applied to minimize the jump of fluxes at interface points, and the effectiveness of this approach is analyzed. Finally, three benchmark problems are solved, and the convergence analysis demonstrates the effectiveness and efficiency of the proposed method. In essence, this paper offers a significant contribution to the literature on fractional differential equations and their numerical solution methodologies. •New approach for solving fractional delay differential equations using spectral element method. Overcomes limitations of traditional methods.•Introduces variable order differentiation matrix and Lagrangian functions for high accuracy. Penalty method minimizes flux jumps at interface points.•Rigorous convergence analysis of proposed method.•Effectiveness demonstrated with practical examples and discretization.