Solving fractal fractional differential equations of a function with respect to another function by using the spectral method

The broad applicability of fractional calculus to modeling physical phenomena via fractional differential equations, along with their complexity, has created substantial demand for efficient analytic and semi-analytic techniques for solving them. This paper has derived a numerical approach to solving a specific class of fractal fractional differential equations (FFDEs) involving the Generalized Caputo-type (GC) of a function with respect to another function, or fractal-$\Psi$-GC, which is shown in this study. The approach is based on an operational matrix (OM) of the fractal fractional derivative of a novel form of orthogonal polynomial. The $\Psi$-normalized shifted Legendre polynomials (NSLP) and $\Psi$-shifted Legendre polynomials (SLP) are introduced. The main characteristic of this approach is that it reduces such problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. Examples are provided to portray the efficiency and applicability of this method. Comparison with similar existing approaches is also conducted to demonstrate the accuracy of the proposed approach.