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This paper investigates a nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. The problem is particularly challenging due to its nonlinear nature, the presence of a time delay, and the incorporation of both fractional diffusion and fractional wave terms, introducing computational complexities for numerical analysis. To address this, we transform the model into a new generalized form involving Riemann–Liouville fractional integrals and a Caputo derivative of order α ∈ (0 , 1). Subsequently, a temporal second-order linearized difference scheme is presented to approximate the model, and its unconditional stability is rigorously proven based on discrete Gronwall's inequality. To reduce computational and storage costs, we extend the discussion to a fast variant of the proposed difference scheme. The sum of exponentials approach is employed to approximate the kernel function in fractional-order operators, leading to fast difference analogs for the Riemann–Liouville fractional integral and the Caputo derivative. A fast variant of the developed direct method is introduced based on these analogs. Numerical results are provided to validate our theoretical findings and assess the accuracy and efficiency of the difference schemes. • Investigation of the nonlinear time-fractional delayed MSDDWE. • A second-order temporal L2-type difference scheme. • Demonstration of the unconditional stability of the numerical scheme. • A fast second-order scheme for the nonlinear time-fractional delayed MSDDWE. • Numerical examples to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α + 1 where 0 < α < 1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L 2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis. [ABSTRACT FROM AUTHOR]