ON A DEGENERATE GENERALIZED INTEGRAL TRANSFORM: PROPERTIES AND APPLICATIONS

Abstract

Abstract

 In this work, we introduce a degenerate generalized integral transform that unifies classical transforms (Laplace, Sumudu, Elzaki, etc.) with their degenerate counterparts. The proposed transform is defined by  

where , and .  In the limit , the classical transforms are recovered, and the degenerate Laplace, Sumudu, and Elzaki transforms are included as special cases. We establish fundamental properties including existence, linearity, differentiation, and convolution theorems. Additionally, we derive transform pairs for elementary functions and special functions such as the Fox-Wright function, the k-Gauss hypergeometric function, and the Mittag-Leffler function. Applications to variable coefficient initial value problems are demonstrated. The novelty of this work lies in the unification of classical and degenerate transforms within a single framework, providing researchers flexibility to select transforms tailored to specific problems while maintaining solution consistency.