This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.

翻译：本文研究一类涉及时间分数阶混合次扩散与扩散波方程（SDDWE）的初边值问题。为便于数值方法与分析，将原问题转化为包含阶数在(0,1)内的Caputo导数与Riemann-Liouville分数阶积分的新型积分-微分模型。通过给出解的先验估计，我们证明了该问题数值解的存在唯一性。提出二阶方法近似分数阶Riemann-Liouville积分，并采用L2型公式近似Caputo导数，由此得到具有时间二阶精度的该模型近似方法。建立了所提差分格式无条件稳定性的证明。此外，我们展示了该方法在构造与分析更广泛的多项时间分数阶混合次扩散与扩散波方程的二阶L2型数值格式方面的潜力。数值结果验证了方法的精度并证实了理论结论。