A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the ψ-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on δψ derivatives—and derive explicit formulas for the transforms of ψ-Riemann–Liouville, ψ-Caputo, and ψ-Hilfer fractional operators. The results provide a rigorous analytical foundation for solving hybrid fractional Cauchy problems that combine classical and fractional derivatives. As an application, we solve a hybrid model incorporating both δψ derivatives and ψ-Hilfer fractional derivatives, obtaining explicit solutions in terms of multivariate Mittag-Leffler functions. The effectiveness of the method is illustrated through a capacitor charging model and a hydraulic door closer system based on a mass-damper model, demonstrating how fractional-order terms capture memory effects and non-ideal behaviors not described by classical integer-order models.