TY - JOUR T1 - On the Construction of The Laplace Transform via Gamma Function AU - Kaya, Ufuk AU - Ermiş, Şeyda PY - 2024 DA - December Y2 - 2024 DO - 10.17798/bitlisfen.1545337 JF - Bitlis Eren Üniversitesi Fen Bilimleri Dergisi PB - Bitlis Eren University WT - DergiPark SN - 2147-3129 SP - 1247 EP - 1259 VL - 13 IS - 4 LA - en AB - The Laplace transform can be applied to integrable and exponential-type functions on the half-line [0,├ ∞)┤by the formula L{f}=∫_0^∞▒〖f(x) e^(-sx) dx〗. This transform reduces differential equations to algebraic equations and solves many non-homogeneous differential equations. However, the Laplace transform cannot be applied to some functions such as x^(- 9/4), because the given integral is divergent. So, the Laplace transform do not solve some differential equations with some terms such as x^(- 9/4). This transform needs a revision to include such functions to solve a wider class of differential equations. In this study, we defined the Ω-Laplace transform, which eliminates such insufficiency of the Laplace transform and is a generalization of it. We applied this new operator to previously unsolved differential equations and obtained solutions. Ω-Laplace ensform given with the help of series:f(x)=∑_(n=0)^∞▒〖c_n x^(r_n ) 〗⇒Ω{f}=∑_(n=0)^∞▒(c_n Γ(r_n+1))/s^(r_n+1) Moreover, we give the similar and different properties of this transform to the Laplace transform. KW - Differential equations KW - Laplace Transform KW - Operators KW - Gamma function KW - Series CR - [1] E. T. Jaynes, Probability Theory: The Logic of Science, vol. 727, Cambridge, UK: Cambridge University Press, 2003. CR - [2] L. Euler, “De constructione aequationum” [The Construction of Equations], Opera Omnia, 1st series (in Latin), vol. 22, pp. 150–161, 1744. CR - [3] J. L. Lagrange, “Mémoire sur l'utilité de la méthode,” Œuvres de Lagrange, vol. 2, pp. 171–234, 1773. CR - [4] I. 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