A novel fitted spline method for the numerical treatment of singularly perturbed differential equations having small delays

This study examines singularly perturbed differential equations with delay on the reaction and convection terms. The considered problem exhibits boundary layer at left end points of the given interval dependent on the convection term sign. We approximated the terms containing the delay using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. A fitting factor is introduced in the highest order derivative term of the singularly perturbed differential equation first, and then, a cubic spline difference scheme is employed for approximating the required solution numerically. The value of fitting factor is obtained through the use of the theory of singular perturbations and Taylor series expansion procedure. The resulting tri-diagonal system is solved with the help of the Thomas algorithm. Convergence of the suggested technique are investigated. Applicability and the computational efficiency of the scheme is demonstrated by solving three example problems. The maximum absolute error of the suggested technique are computed and the computational results are compared with the results obtained by other methods. The effect of shifts on the solutions is investigated by depicting the figures.