Fitted operator method over Gaussian quadrature formula for parabolic singularly perturbed convection-diffusion problem

In this manuscript, a new exponentially fitted operator strategy for solving a singularly perturbed parabolic partial differential equation with a right boundary layer is considered. We discretize the time variable using the implicit Euler approach and approximate the equation into first order delay differential equation with a small deviating argument using a Taylor series expansion. The two-point Gaussian quadrature formula and linear interpolation are implemented to obtain a tridiagonal system of equations. The tridiagonal system of equations is solved using the Thomas algorithm. Three numerical examples are considered to illustrate the efficiency of the present method and compared with the methods produced by different authors. Convergence of the method is analyzed. The absolute maximum error and rate of convergence are obtained for the model examples. The result shows that the present method is more accurate and $\epsilon$-uniformly convergent for all $\epsilon\leqslant h$.