Solving a singular beam equation by using a weak-form integral equation method

Solving second-order singularly perturbed ODE by the collocation method based on energetic Robin boundary functions

“…Therefore, ϕ j (x) is a normalized exponential trial function. Liu et al [18] extended the above trial functions in the weak-form formulation of the fourthorder singular beam equation to find the numerical solution.…”

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

“…Dong et al have focused on the different weak‐form formulations and the ill‐posed properties for the fourth‐order ODEs with various boundary conditions. Liu et al have proposed a weak‐form integral equation method with exponentially and polynomially fitted trial solutions, and they are designed to satisfy automatically the boundary conditions. That paper found accurate numerical solutions of singular beam equations.…”

“…In our former works, the idea of using weak‐form integral equations together with different test functions and trial functions has been successfully used in solving ODEs, for example, in previous literature . In this paper, we are going to extend the idea by using the sinusoidal functions as test functions and also the bases, including the boundary functions, to develop a very powerful beam solver for a nonlinear beam equation with an integral term of the deformation energy.…”