A taylor series-based iterative algorithm for fourth-order parabolic partial differential equations arising in beam vibration analysis

This study presents a novel developed approach (NDA) for obtaining approximate analytical solutions of fourth-order parabolic partial differential equations arising in the transverse vibration analysis of homogeneous beams. The proposed method employs Taylor series expansion within an efficient iterative framework to construct rapidly convergent series solutions. Three numerical applications are considered to demonstrate the effectiveness and reliability of the proposed algorithm. The approximate solutions obtained using NDA are compared with the corresponding exact solutions to assess the accuracy and efficiency of the proposed approach. Furthermore, absolute error analysis and two-dimensional graphical representations are provided to evaluate the performance of the proposed technique over the computational domain. The numerical results demonstrate that the NDA solutions are in excellent agreement with the exact solutions. The convergence analysis further indicates that only a few iterations are required to obtain highly accurate approximations.