arXiv:1907.04541v1 [math.CA] 10 Jul 2019 On Ψ-Laplace transform method and its applications to Ψ-fractional differential equations Hafiz Muhammad Fahad a,∗ , Mujeeb ur Rehman a, a Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad Pakistan Abstract Motivated by some recent developments in Ψ-fractional calculus, in this paper some new properties and the uniqueness of Ψ-Laplace transform in the settings of Ψ-fractional calculus are established. The final goal of this research is to demonstrate the effectiveness of Ψ-Laplace transform for solving Ψ-fractional ordinary and partial differential equations. Keywords: Fractional calculus; generalized Laplace transform; generalized fractional operators; Ψ-convolution structure 1. Introduction The birth of fractional calculus finds its roots in the last years of the seventeenth century, when Newtons work along with Leibnizs served basis for the inception of classical calculus. Leibniz devised the notation d n dx n f (x) to denote the nth-order derivative of the function f . When he communicated this to de l’Hospital, the later asked what the meaning of the said notation would be if n = 1 2 . This communication in the present day is unanimously considered to be the foundation of fractional calculus. At present, this field has become a matter of deep interest for many researchers. In fractional calculus, the Riemann-Liouville fractional derivative is note-worthy but it has certain disadvantages when trying to model physical problems because of its inappro- priate physical conditions. Caputo made a significant contribution by affirming the defini- tion of fractional derivative which is suitable for physical conditions [2]. Moreover, several other families of fractional operators have been introduced until now, out of which Liouville, * Corresponding author. Tel +92 333 6330824 Email addresses: hafizmuhammadfahad13@gmail.com ( Hafiz Muhammad Fahad), mujeeburrehman345@yahoo.com ( Mujeeb ur Rehman) Preprint submitted to Elsevier July 11, 2019