Cubic non-polynomial spline on piecewise mesh for singularly perturbed reaction differential equations with robin type boundary conditions

The solution domain is first discretized using a piecewise mesh. The process begins by defining the cubic non-polynomial spline function and calculating its derivatives. These derivatives are then transformed into difference approximations, forming a linear system of algebraic equations in the form of a three-term recurrence relation, which is solved using an elimination algorithm. The stability and consistency of the method are analyzed, ensuring convergence. Numerical model examples are used to validate the proposed method, and the results are compared with those from other methods found in the literature. The maximum absolute error and the order of convergence for each example demonstrate the effectiveness and core contribution of the method.