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Anomalous relaxation and diffusion processes have been widely characterizedby fractional derivative models, where the definition of the fractional-orderderivative remains a historical debate due to the singular memory kernel thatchallenges numerical calculations. This study first explores physicalproperties of relaxation and diffusion models where the fractional derivativewas defined recently using an exponential kernel. Analytical analysis showsthat the fractional derivative model with an exponential kernel cannotcharacterize non-exponential dynamics well-documented in anomalous relaxationand diffusion. A legitimate extension of the previous fractional derivative isthen proposed by replacing the exponential kernel with a stretched exponentialkernel. Numerical tests show that the fractional derivative model with thestretched exponential kernel can describe a much wider range of anomalousdiffusion than the exponential kernel, implying the potential applicability ofthe new fractional derivative in quantifying real-world, anomalous relaxationand diffusion processes.

Fractional derivative defined by non-singular kernels to capture anomalous relaxation and diffusion