The Gbenga Gideon Integral Transform: A Four-Parameter Unifying Framework for Laplace-Type Transforms, with Rigorous Theory, Engineering Applications, and Numerical Validation

Abstract

Over the past three decades, a large family of Laplace-type integral transforms has been proposed, including the Sumudu, Elzaki, Aboodh, Mohand, Mahgoub, Kamal, Shehu, Sadik, Sawi, and Natural transforms, each defined by a kernel of the form 𝜑 𝑣 𝑒 ;𝜓 𝑣 𝑡 and each accompanied by its own operational calculus. The proliferation of these transforms has produced a fragmented literature in which essentially equivalent operational results are repeatedly rederived for each new kernel. This study presents the Gbenga Gideon (G.G.) integral transform, a four-parameter generalized integral transform defined by 𝒢{𝑓 𝑡 } 𝑣 = 𝑝 𝑣 𝑚 𝑒 ;𝑞𝑣 𝑤 ∞ 𝑡 0 𝑓 𝑡 𝑑𝑡, and establishes it as a single unifying framework from which the operational calculus of at least thirty-eight published transforms is recovered by parameter specialization. The transform is developed axiomatically. Existence and convergence are established for piecewise continuous functions of exponential order, and the fundamental operational properties, linearity, shifting, change of scale, transforms of derivatives, multiplication by powers of 𝑡, convolution, inversion, and uniqueness, are proved as formal theorems. The framework is then applied to canonical initial value problems in electrical circuit analysis, heat conduction and cooling, population growth, radioactive decay, and mechanical vibrations. Nine theorems with complete proofs are presented, together with a parameter dictionary mapping the G.G. integral transform onto existing transforms. Analytical solutions obtained through the G.G. integral transform is validated against high-accuracy Runge-Kutta integration: across eight benchmark problems the maximum absolute discrepancy does not exceed 2.3 × 10;7 and root-mean-square errors lie between 10;11 and 10;8 , confirming exactness to numerical precision. A symbolic benchmark shows the transform-domain route resolving a damped oscillator initial value problem approximately 3.4 times faster than direct symbolic integration. The G.G. integral transform provides a mathematically rigorous, computationally efficient, and pedagogically economical umbrella for the Laplace-type transform family: a property proved once for the G.G. integral transform holds automatically for every member obtained by specialization.