Derivation of a 3-Step Numerical Method for Stiff Differential Equation Using the Hermite Polynomial as a Basic Function

This study evaluates the three-step first derivative block hybrid method with six intra-points (TSDBHM6) for solving stiff differential equations, comparing its performance with existing methods from the literature. Stiff problems are characterized by solutions where certain components decay significantly faster than others, typically with eigenvalues having negative real parts. The TSDBHM6 method provides numerical solutions over non-overlapping intervals without needing initial values from alternative methods, as initial conditions are explicitly computed at each stage. Two stiff problems are analyzed: a linear problem with the exact solution x(t) = sin(t) +〖 e〗^(-20t) and a nonlinear highly stiff problem with the exact solution x(t) = √(1+e^(-100t) ). Numerical results for these problems are compared with the solutions obtained using 2-point Implicit Block Method with an Off-stage Function (2P4BBDF), Fifth Order Block Backward Differentiation Formula (BBDF(5)), and Improved 2-point Block Backward Differentiation Formula of order five (I2BBDF(5)) from the literature. Findings show that TSDBHM6 provides numerical solutions consistent with exact solutions and achieves lower maximum errors than the methods from the literature across various step sizes (〖10〗^(-3),〖10〗^(-5),〖10〗^(-7) )The results demonstrate that TSDBHM6 is a robust and accurate method for handling stiff problems, outperforming existing approaches and offering a valuable tool for practical applications in solving stiff differential equations.