Solution of Sakata-Taketani Equation via the Caputo and Riemann-Liouville Fractional Derivatives

In this paper, we present the Sakata–Taketani (ST) equation of spin 0 in the framework of fractional derivatives. We propose fractional Lagrangian and Hamiltonian densities. Using the fractional variational principle, the fractional Sakata–Taketani equation and its adjoint are determined. Accordingly, we have solved exactly the following problems: the free Sakata-Taketani equation of spin 0 in both cases (in time fractional and in space-time fractional cases) and the Sakata–Taketani particle of spin 0 in space-time fractional and in a box model (also known as the infinite potential well).