مشخصات پژوهش

Fully fourth-order nonlinear boundary value problems are essential for modeling a variety of phenomena in engineering and the applied sciences, such as fluid dynamics, quantum mechanics, reaction-diffusion processes, and beam and plate theories in elasticity. However, because they are nonlinear and higher-order, analytical solutions are unfeasible, and current numerical methods frequently encounter stability, accuracy, and efficiency problems. The dense discretization typically required by traditional low-order schemes results in high computational costs and error accumulation, particularly when handling boundary conditions. This makes high-order numerical techniques that can produce precise answers with less computing power imperative. An effective improvement that strengthens the overall performance of the numerical method is the use of corrected trapezoidal quadrature, which increases the accuracy of integral approximations. The significance of this study resides in creating a sound, effective, and precise framework that not only addresses the drawbacks of traditional methods but also expands the use of numerical mathematics to include intricate nonlinear fourth-order models found in contemporary science and engineering.