On the Construction of The Laplace Transform via Gamma Function

Year 2024, Volume: 13 Issue: 4, 1247 - 1259, 31.12.2024

Ufuk Kaya , Şeyda Ermiş

https://doi.org/10.17798/bitlisfen.1545337

Abstract

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.

Keywords

Differential equations, Laplace Transform, Operators, Gamma function, Series

Ethical Statement

The study is complied with research and publication ethics.

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