Solving singular convection-diffusion equation by exponentially-fitted trial functions and adjoint Trefftz test functions

The paper develops a weak-form integral equation method (WFIEM) for solving the singularly perturbed convection-diffusion equation, which is too ill-posed to find the singular solution using conventional methods. We use Green's second identity to generate integral equation, which includes a source term and boundary functions on the space-time boundary, and the derived adjoint Trefftz test functions. Then the singular solution is expressed in terms of a set of exponentially-fitted trial functions, which automatically satisfy the boundary conditions. The numerical algorithm deduced from the WFIEM is effective and accurate in the numerical solutions of highly singular parabolic type problems as will be observed by numerical examples.