High-Order Accurate Differential-Difference Schemes on a Minimal Stencil for Euler–Bernoulli Beam Equation with Second Dirichlet Boundary Value Condition

In this article, we propose a method for solving the initial boundary value problem (IBVP) associated with a fourth-order partial differential equation (PDE), specifically the Euler–Bernoulli beam equation. Our approach is based on discretizing the spatial variable using the method of moments with a carefully selected set of basic functions. By introducing auxiliary functions and performing interpolation and integration over the spatial variable, we derive a system of second-order ordinary differential equations (ODEs) from the original equation. To accommodate internal and near-boundary nodes, we employ Newton–Stirling and Hermite–Birkhoff interpolations. Importantly, our method automatically satisfies the boundary conditions, eliminating the need for separate approximations as typically required in classical numerical methods. The primary objective of this work is to construct high-order accurate differential-difference schemes with minimal stencils that approximate fourth-order parabolic equations under various types of boundary conditions. We establish the convergence of these schemes using the logarithmic norm of the matrix. To validate our theoretical findings, we conduct numerical experiments that confirm the attained results.