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 \(g'\). This attribute leads to the investigation of Caputo fractional derivative of \(\alpha \)-fractal splines rather than just a continuous non-differentiable \(\alpha \)-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.