Geometric Interpretation for Riemann–Liouville Fractional-Order Integral

A new method is proposed to plot the image of Riemann-Liouville (RL) fractional-order integral. The meanings of the image are discussed, including the mathematical expression of the image, the corresponding relationship between the image and RL fractional-order integral, and the change of the image as increasing the upper limit of RL fractional-order integral. The image is likened to a process of light refraction for interpreting the geometric meanings of RL fractional-order integral. The exponential and duality properties of RL fractional-order integral are interpreted by the image. The duality property shows that there is no essential difference between integer-order and RL fractional-order integrals, except different observation angles. It concludes that RL fractional-order integral is a projection of a line integral on a plane.