Fourth-Order Numerical Derivation as Being an Inverse Force Problem of Beam Equations

Besides the closed-form expansion coefficients of a weak-form numerical differentiator (WFND), we introduce a cubic boundary shape function with the aid of two parameters for reducing the boundary errors of fourth-order numerical derivatives to zero. So that the accuracy of numerical derivatives obtained by the new WFND can be improved significantly. The fourth-order numerical derivation can be modeled as a linear beam equation subjecting to specified boundary conditions and displacements to recover an unknown forcing term. By means of boundary shape functions, two numerical collocation methods automatically satisfying the boundary conditions are developed. For a simply supported linear Euler–Bernoulli beam with an elastic foundation, the unknown spatially–temporally dependent force is retrieved. The displacement at a final time and strain on the right-boundary of the beam are over-specified to recover the external force using the method of superposition of boundary shape functions (MSBSF). When the displacement is determined to satisfy the prescribed right-boundary strain, we can recover an unknown spatially–temporally dependent force by inserting the displacement into the linear beam equation. An embedded method (EM) is developed to transform the linear beam model into a vibrating linear beam equation, and then we can develop a robust technique to compute the fourth-order derivative of noisy data by using the EM and MSBSF. The four proposed methods for evaluating the fourth-order derivatives of noisy data are efficient and accurate.