Abstract:
This thesis presents an accurate computational approach utilizing piecewise mesh for solving
singularly perturbed differential equations with mixed shifts. The proposed method begins by
discretizing the solution domain using a defined piecewise mesh size. The singularly perturbed
differential problems are first transformed into asymptotically equivalent singularly perturbed
boundary value problems. Subsequently, the finite difference approximation technique is
employed to convert the derivatives in the differential equations into a linear system of algebraic
equations, represented as a three-term recurrence relation that can be effectively solved using
the Thomas algorithm. The stability and consistency of the method are rigorously investigated,
ensuring the convergence of the proposed approach. Additionally, the influence of delay and
advance parameters on the solution profile is explored. To validate the method's applicability,
various examples are solved for different perturbation parameters and mesh sizes. The
numerical results obtained are compared with existing findings in the literature, demonstrating
that the proposed method significantly enhances accuracy. Furthermore, graphical
representations illustrate the solution's behavior concerning varying shifts and boundary layer
dynamics.
