On The Construction of the Laplace Transform via Gamma Function

Abstract

The Laplace transform can be applied to integrable and exponential-type functions on the half-line [0,∞) by the formula 𝐿�{𝑓�} = ∫ 𝑓�(𝑥�)𝑒�−𝑠�𝑥�𝑑�𝑥� ∞ 0 . This transform reduces differential equations to algebraic equations and solves many nonhomogeneous 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 can not solve some differential equations with some terms such as 𝑥�− 9 4 . This transform requires revision to accommodate such functions and 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 transform given with the help of series: ∞ 𝑓�(𝑥�) = ∑𝑐�𝑛�𝑥�𝑟�𝑛� 𝑛�=0 ∞ ⇒Ω{𝑓�} = ∑𝑐�𝑛�Γ(𝑟�𝑛� +1) 𝑠�𝑟�𝑛�+1 𝑛�=0 Moreover, we compare the similarities and differences of this transform with the Laplace transform.