No, in general, an IVP/BVP with a different fractional derivative will not give the same exact solution, since there are several definitions of fractional derivatives, including Riemann-Liouville, Caputo, ...etc. These definitions capture different aspects of what it means to take a fractional power of a derivative. Each fractional derivative definition has its own properties regarding integration and differentiation. These properties affect how the solution behaves within the equation. Therefore, depending on the specific fractional derivatives used and the chosen initial/boundary conditions, the solutions to the IVP/BVP can differ significantly.

In some special cases, with specific choices of functions and conditions, solutions from different fractional derivatives might coincide. However, this is not the general rule.

Ilhem Kadri Thanks a lot for your sensible reply. However, I found some people develop numerical methods for FDE with conformable derivative, say, but they do the numerical experiments for the problems which have exact solutions in the sense of other fractional derivatives. And finally, they also compare their methods with the methods which are developed for non-conformable FDE.

Is this logical?

The logic behind what you described has some validity, but also raises questions.

Comparing the new method with methods for non-conformable FDEs might not be directly relevant. The conformable derivative has its own properties, and methods designed for other types might not translate well.

IVP/BVP with different fractional derivatives doesn't give the same exact solution because each fractional derivative has its own perculiar properties different from the other. For instance, the properties of the Caputo fractional derivative is different from that of the Riemann Liouvillederivative