The Gbenga Gideon Integral Transform and Its Application to Second-Order Ordinary Differential Equations

This study develops the Gbenga Gideon (G.G.) integral transform, a four-parameter generalization of the Laplace transform characterized by the kernel (p v^m e^{-qv^w t}), and establishes its operational calculus for solving second-order ordinary differential equations with constant coefficients. Fundamental transform pairs are derived for powers, exponential, trigonometric, and hyperbolic functions, while key operational properties, including linearity, first and second shifting, change of scale, differentiation, multiplication by the independent variable, and convolution, are formulated and proved. Particular attention is given to the corrected general change-of-scale formula and a convolution theorem that is consistent with the transform’s four-parameter normalization. The parameter values under which the G.G. integral transform reduces to the Aboodh, Gupta, Kamal, Elzaki, and Sadik transforms are systematically identified, tabulated, and verified against the established definitions of these transforms. The applicability of the proposed method is demonstrated through nine second-order initial- and boundary-value problems involving homogeneous equations with real, repeated, and complex characteristic roots, as well as nonhomogeneous equations with exponential, polynomial, and damped-trigonometric forcing terms. General convolution solutions are also obtained for the equations (y''-a^2y=f(t)) and (y''+a^2y=f(t)). The resulting solutions are cross-validated using the Laplace, Aboodh, and Elzaki transforms and further confirmed through direct substitution into the corresponding differential equations. The results demonstrate the consistency and versatility of the G.G. integral transform as an operational method for solving second-order ordinary differential equations.