612b1774-f55c-466c-b429-5011d8154e1e20220509091948947wseas:wseasmdt@crossref.orgMDT DepositEQUATIONS2732-99762944-914610.37394/232021https://wseas.com/journals/equations/32020223202022210.37394/232021.2022.2https://wseas.com/journals/equations/2022.phpD. C.RoachDepartment of Engineering University of New Brunswick 100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5, CANADAM. H.HamdanDepartment of Engineering University of New Brunswick 100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5, CANADAIn this work, the problem of obtaining particular and general solutions to Airy’s inhomogeneous equation when the forcing function is one of Einstein’s functions is examined. Expressions for the particular solutions provide connections between the Nield-Kuznetsov and Einstein functions. Computations have been carried out using Wolfram Alpha.59202259202248538https://wseas.com/journals/equations/2022/a16equations-008(2022).pdf10.37394/232021.2022.2.8https://wseas.com/journals/equations/2022/a16equations-008(2022).pdfAbramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover, New York, 1984. Khanmamedov, A.Kh., Makhmudova1, M.G. and Gafarova, N.F., Special Solutions of the Stark Equation, Advanced Mathematical Models & Applications, Vol. 6(1), 2021, pp. 59-62. Temme, N.M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996. Hilsenrath, J. and Ziegler, G.G., Tables of Einstein Functions, Nat. Bur. Stand. (U.S.), Monograph 49, 258 pages, 1962. 10.6028/jres.074b.016Cezairliyan, A., Derivatives of the Grüneisen and Einstein Functions, Journal of Research of the National Bureau of Standards- B. Mathematical Sciences, Vol. 74B #3, 1970, pp. 175-182. Wolfram Alpha, www.mathworld.wolfram.com. 10.1098/rspa.2003.1156Maximon, L.C. The Dilogarithm Function for Complex Argument, Proc. R. Soc. Lond. A, Vol. 459, 2003, pp. 2807–2819. Lewin, L., ed., Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, Vol. 37, 1991, Providence, RI: Amer. Math. Soc. ISBN 978-0-8218-1634-9. 10.37394/232011.2022.17.5Roach, D.C. and Hamdan, M.H., On the Sigmoid Function as a Variable Permeability Model for Brinkman Equation, Trans. on Applied and Theoretical Mechanics, WSEAS, Vol. 17, 2022, pp. 29-38. 10.37394/232020.2022.2.13Hamdan, M.H. and Roach, D.C., The Sigmoid Neural Network Activation Function and its Connections to Airy’s and the Nield-Kuznetsov Functions, Proof, WSEAS, Vol. 2, 2022, pp. 108-114. Airy, G.B., On the Intensity of Light in the Neighbourhood of a Caustic, Trans. Cambridge Phil. Soc., Vol. 6, 1838, pp. 379-401. Vallée, O. and Soares, M., Airy functions and applications to Physics. World Scientific, London, 2004. 10.1016/j.amc.2011.02.025Hamdan, M.H. and Kamel, M.T., On the Ni(x) Integral Function and its Application to the Airy's Non-homogeneous Equation, Applied Math. Comput. Vol. 21(17), 2011, pp. 7349-7360. Miller, J. C. P. and Mursi, Z., Notes on the solution of the equation y ′′ − xy = f(x), Quarterly J. Mech. Appl. Math., Vol. 3, 1950, pp. 113-118. 10.1007/s11242-009-9342-0Nield, D.A. and Kuznetsov, A.V., The effect of a transition layer between a fluid and a porous medium: shear flow in a channel, Transp Porous Med, Vol. 78, 2009, pp. 477-487. Jayyousi Dajani, S. and Hamdan, M.H., Airy’s Inhomogeneous Equation with Special Forcing Function, İSTANBUL International Modern Scientific Research Congress –II, Istanbul, TURKEY, Proceedings, ISBN: 978-625-7898-59-1, IKSAD Publishing House, Dec. 23-25, 2021, pp. 1367-1375. 10.1137/21m1401590Dunster, T.M., Nield-Kuzenetsov Functions and Laplace Transforms of Parabolic Cylinder Functions, SIAM J. Math. Anal., Vol. 53(5), 2021, pp. 915- 5947.