ISSN (Online): 3108-1584
Volume 2, Issue 2 - September 2026
This study presents a four-parameter generalization of the Laplace transform, here called the Gbenga Gideon (G.G.) integral transform (GGIT), with kernel pv^m 〖{e〗^(-qv^w t)}, and develops its operational calculus for the solution of third-order ordinary differential equations. Elementary transform pairs are derived for constant, power, exponential, trigonometric and hyperbolic functions, and the fundamental operational properties — linearity, first and second shifting, change of scale, differentiation, multiplication by the independent variable, Laplace–GGIT duality and convolution — are stated and proved. A compact formula for the transform of the n-th derivative is established, with the third-order specialization required for third-order initial-value problems singled out. The parameter choices through which the G.G. kernel reduces to thirty-eight named transforms from the recent literature, including the Laplace, Sumudu, Aboodh, Elzaki, Sawi, Shehu and Sadik transforms, are tabulated. The method is then applied to third-order constant-coefficient equations with trigonometric and exponential-polynomial forcing terms. The GG Integrals Transform solutions were bench-marked against key existing transforms (Laplace, Aboodh, and Elzaki) and the results were shown to agree.
G.G. integral transform, third-order ordinary differential equation, integral transform, Laplace transform, Aboodh transform, Elzaki transform, analytical solution
Abraham D.A., Oyetoro G. G., Afolabi O.A., Oyefusi A.S., "The Gbenga Gideon Integral Transform and Its Application to Third-Order Ordinary Differential Equations", Cosmo Research & Science International Journal, vol. Jul-25, no. 1, pp. 33-49, 2026.
Abraham D.A., Oyetoro G. G., Afolabi O.A., Oyefusi A.S. (2026). The Gbenga Gideon Integral Transform and Its Application to Third-Order Ordinary Differential Equations. Cosmo Research & Science International Journal, Jul-25(1), 33-49.
Abraham D.A., Oyetoro G. G., Afolabi O.A., Oyefusi A.S.. "The Gbenga Gideon Integral Transform and Its Application to Third-Order Ordinary Differential Equations." Cosmo Research & Science International Journal, vol. Jul-25, no. 1, 2026, pp. 33-49.
@article{CRSIJ26000344,
author = {Abraham D.A., Oyetoro G. G., Afolabi O.A., Oyefusi A.S.},
title = {The Gbenga Gideon Integral Transform and Its Application to Third-Order Ordinary Differential Equations},
journal = {Cosmo Research and Science International Journal},
year = {2025},
volume = {2},
number = {2},
pages = {33-49},
issn = {3108-1584},
url = {https://cosmorsij.com/published/CRSIJ26000344.pdf},
abstract = {This study presents a four-parameter generalization of the Laplace transform, here called the Gbenga Gideon (G.G.) integral transform (GGIT), with kernel pv^m 〖{e〗^(-qv^w t)}, and develops its operational calculus for the solution of third-order ordinary differential equations. Elementary transform pairs are derived for constant, power, exponential, trigonometric and hyperbolic functions, and the fundamental operational properties — linearity, first and second shifting, change of scale, differentiation, multiplication by the independent variable, Laplace–GGIT duality and convolution — are stated and proved. A compact formula for the transform of the n-th derivative is established, with the third-order specialization required for third-order initial-value problems singled out. The parameter choices through which the G.G. kernel reduces to thirty-eight named transforms from the recent literature, including the Laplace, Sumudu, Aboodh, Elzaki, Sawi, Shehu and Sadik transforms, are tabulated. The method is then applied to third-order constant-coefficient equations with trigonometric and exponential-polynomial forcing terms. The GG Integrals Transform solutions were bench-marked against key existing transforms (Laplace, Aboodh, and Elzaki) and the results were shown to agree.},
keywords = {G.G. integral transform, third-order ordinary differential equation, integral transform, Laplace transform, Aboodh transform, Elzaki transform, analytical solution},
month = {September}
}