$\psi$–Katugampola Fractional Derivatives and
Integrals-Application to Mass–Spring Damper System
Abstract
We propose a new type of generalized fractional derivatives with respect
to (wrt) another function. These new generalized fractional derivatives
generalize $\psi$–Caputo, Riemann–Liouville (R–L)
wrt another function, Caputo Hadamard wrt another function, R–L
Hadamard wrt another function, Caputo, R–L, Caputo Hadamard and R–L
Hadamard fractional derivatives. We propose a newly modified Laplace
transform for linear $\psi$–Katugampola fractional
differential equations (FDEs). Properties of this newly generalized
Laplace transform are analyzed. Cauchy problems and mass-spring damper
system with $\psi$–Katugampola fractional derivative
are solved analytically by means of modified Laplace transform. Finally,
a new numerical method is proposed for nonlinear
$\psi$–Katugampola FDEs.