Solving singularly perturbed problems by a weak-form integral equation with exponential trial functions

The second-order singularly perturbed problem is transformed to a singularly perturbed parabolic type partial differential equation by using a fictitious time technique. Then we use Green's second identity to derive a boundary integral equation in terms of the adjoint Trefftz test functions, namely a weak-form integral equation method (WFIEM). It accompanying with the exponential trial functions, which are designed to satisfy the boundary conditions automatically, can provide very accurate numerical solutions of linear and nonlinear singularly perturbed problems. For the latter problem the iterative procedure is convergent very fast.