  This paper is in concern with Cauchy problems involving the fractional
derivatives with respect to another function. Results of existence, uniqueness,
and Taylor series among others are established in appropriate functional
spaces. We prove that these results are valid at once for several standard
fractional operators such as the Riemann-Liouville and Caputo operators, the
Hadamard operators, the Erd\'elyi-Kober operators, etc., depending on the
choice of the scaling function. We also show that our technique can be useful
to solve a wide range of Volterra integral equations. The numerical
approximation of solutions of systems involving the fractional derivatives with
respect to another function is also investigated and the optimal convergence
rate of the schemes is reached in graded meshes, even in the case of singular
solutions. Various examples and numerical tests, with an application to the
Erd\'elyi-Kober operators, are performed at the end to illustrate the
efficiency of the proposed approach.
