Solving fractal-fractional differential equations using operational matrix of derivatives via Hilfer fractal-fractional derivative sense

This study will introduce a new differentiation operator, the Hilfer fractional-fractal derivative (H-FFD). The new proposed derivative aims to attract more non-local problems that show with the same time fractal behaviors. For numerical settlement of initial value problems, we use the shifted Legendre operational matrix. The main advantage of this method is that it reduces both linear and non-linear problems alike in solving the problem into a system of linear and non-linear algebraic equations. In addition, the numerical approximation of this new operator also offers some applications to systems of linear and non-linear problems.