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

Choden, Samten; Sompong, Jakgrit; Thailert, Ekkarath; Ntouyas, Sotiris K.

doi:10.3390/fractalfract9110751

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

¹

Department of Mathematics, Naresuan University, Phitsanulok 65000, Thailand

²

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

^*

Author to whom correspondence should be addressed.

Fractal Fract. 2025, 9(11), 751; https://doi.org/10.3390/fractalfract9110751

Submission received: 13 October 2025 / Revised: 14 November 2025 / Accepted: 18 November 2025 / Published: 20 November 2025

(This article belongs to the Special Issue From Fractals to Fractional Calculus: Nonlinear Dynamics and Hybrid Complex Systems)

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Abstract

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

δ_{ψ}

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

δ_{ψ}

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.

Keywords: generalized Laplace transform; ψ-Hilfer fractional derivative; δψ derivatives; hybrid fractional Cauchy problems; weighted function spaces; fractional differential equations

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MDPI and ACS Style

Choden, S.; Sompong, J.; Thailert, E.; Ntouyas, S.K. A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems. Fractal Fract. 2025, 9, 751. https://doi.org/10.3390/fractalfract9110751

AMA Style

Choden S, Sompong J, Thailert E, Ntouyas SK. A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems. Fractal and Fractional. 2025; 9(11):751. https://doi.org/10.3390/fractalfract9110751

Chicago/Turabian Style

Choden, Samten, Jakgrit Sompong, Ekkarath Thailert, and Sotiris K. Ntouyas. 2025. "A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems" Fractal and Fractional 9, no. 11: 751. https://doi.org/10.3390/fractalfract9110751

APA Style

Choden, S., Sompong, J., Thailert, E., & Ntouyas, S. K. (2025). A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems. Fractal and Fractional, 9(11), 751. https://doi.org/10.3390/fractalfract9110751

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

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