This work considers the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Rigorous mathematical and numerical analysis for this model is not available in the literature, partly because the variable-exponent Abel kernel may not be positive definite or monotonic. We overcome these difficulties to design two numerical schemes and derive their stability and error estimate based on the proved solution regularity, with $\alpha(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings.

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