Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

Abstract

For the third-order linearly singularly perturbed problems under four different types boundary conditions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. In the WFIEM the exponentially and polynomially fitted trial functions are designed to satisfy the boundary conditions automatically, while the test functions satisfy the adjoint boundary conditions exactly. The WFIEM provides accurate and stable solutions to the highly singular third-order problems.

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

“…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.…”