ACCURATE NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC DELAY DIFFERENTIAL EQUATION

<strong>Abstract:</strong> Numerical treatment of the singularly perturbed parabolic delay differential equation is a challenging task due to the presence of a boundary layer, which can make it difficult to accurately compute the solution. In this context, an accurate numerical scheme is proposed using the θ-method for time discretization and a non-standard finite difference method for space discretization for the singularly perturbed parabolic delay differential equation. This approach allows for a more accurate representation of the solution near the bounder layer,where the behavior of the solution can change rapidly. The stability and uniform convergence of the proposed numerical scheme are investigated. It is shown that the scheme is uniformly convergent with first order convergence before applying Richardson extrapolation. After applying Richardson extrapolation, the scheme achieves second order convergence. The example involves solving the singularly perturbed parabolic delay differential equation with given coefficients and source term. In conclusion, the numerical treatment of the singularly perturbed parabolic delay differential equation requires careful consideration of the bounder layer and the choice of numerical methods. The theoretical analysis demonstrates the stability and uniform convergence of the scheme, while the numerical example validates its performance.