Semi-analytical solutions of higher-order BVPs used in modeling hydrodynamic stability, material science, astrophysics, and high-order multi-layer structures

Almost all mechanical, physical, or biological processes have been implemented using differential equations (DEs). Many branches of physical sciences and engineering use higher-order ordinary differential equations (ODEs) to simulate complicated systems, climate and environmental models, fluid dynamics, stability analysis, and phenomena. Because of their nonlinearity or difficult boundary conditions, analytical solutions to these equations are sometimes unattainable. For handling 14th-order ODEs, this research looks at how two well-known semi-analytical methods, the Adomian decomposition method (ADM) and the differential transform method (DTM), are both used. Part of this study involves testing the techniques on linear and nonlinear problems. The DTM transforms differential equations into algebraic recurrence relations, enabling efficient series solutions. The ADM decomposes nonlinear terms into Adomian polynomials, allowing iterative solutions without linearization or perturbation assumptions. Across all problems, the ADM and DTM methods consistently achieved high accuracy, with absolute errors ranging from as low as $ 10^{-10} $ to a maximum of $ 7.17 \times 10^{-5} $. The smallest errors ($ 10^{-10} $ to $ 10^{-9} $) occurred for linear problems with exponential and trigonometric exact solutions, while the largest errors ($ 10^{-5} $) appeared in nonlinear problems at $ \xi = 1.0 $. Compared with the Haar wavelet, the improved residual power series method, the spline methods, and the homotopy perturbation method/optimal homotopy asymptotic method, our ADM and DTM demonstrate superior accuracy and convergence for 14th-order boundary value problems. Lastly, the convergence analysis is described in accordance with the typical theoretical results.