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As we are aware, various types of methods have been proposed to approximatethe Caputo fractional derivative numerically. A common challenge of the methodsis the non-local property of the Caputo fractional derivative which leads tothe slow and memory consuming methods. Diffusive representation of fractionalderivative is an efficient tool to overcome the mentioned challenge. This paperpresents two new diffusive representations to approximate the Caputo fractionalderivative of order $0<\alpha<1$. Error analysis of the newly presented methodstogether with some numerical examples are provided at the end.

The sine and cosine diffusive representations for the Caputo fractional derivative