Generalized Laplace Transform for Higher-Order Hybrid Fractional Cauchy Problems: Theory and Applications to Memory-Dependent Dynamics

This paper develops a generalized Laplace transform framework on weighted function spaces Cδψ,γn[a,b], establishing a symmetry between integer-order δψ operators and fractional ψ-Hilfer derivatives at the level of transform representations. Explicit transformation formulas are derived for the nth-order δψ-derivative and the ψ-Hilfer fractional derivative of order α∈(m−1,m), with m≤n. These results form an analytical basis for the treatment of higher-order hybrid fractional Cauchy problems that systematically couple integer-order and fractional operators subject to mixed initial conditions. The general solution is expressed in closed form using a bivariate Mittag–Leffler function. To illustrate the utility of the approach, a representative second-order hybrid model is studied and compared numerically with its classical integer-order counterpart. The simulations reveal significant differences in the dynamical response, including variations in amplitude, damping behavior, and long-term evolution.