Caputo fractional derivative of \(\alpha \)-fractal spline

The Caputo fractional derivative of a real continuous function g distinguishes from the other fractional derivative methods with the demand for the existence of its first order derivative \(g'\) $g^{\prime }$ . This attribute leads to the investigation of Caputo fractional derivative of \(\alpha \) $\alpha$ -fractal splines rather than just a continuous non-differentiable \(\alpha \) $\alpha$ -fractal function. A bounded linear operator corresponding to the Caputo fractional derivative of fractal version is reported. In addition, a new family of fractal perturbations is proposed in association with the fractional derivative. Thereafter, a numerical approach is used to determine the exact Caputo fractional derivative of fractal functions in terms of Legendre polynomials.