DERIVATION AND IMPLEMENTATION OF LINEAR BLOCK ALGORITHM FOR THE NUMERICAL SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATIONS

This paper presents a new continuous block algorithm for solving first-order initial value problems of ordinary differential equations (ODEs). The block algorithm offers a continuous approximation that enhances accuracy in obtaining the numerical solution of ODEs. The method is developed based on interpolation and collocation techniques, yielding a continuous block formula that can solve ODEs directly. The stability and convergence properties of the method are analyzed, demonstrating its reliability and efficiency for solving ODEs. Numerical experiments are performed to assess the accuracy and effectiveness of the proposed algorithm, comparing it with existing methods. The results indicate that the new block algorithm performs well, providing more accurate solutions and improved efficiency. This approach proves highly effective in addressing both stiff and non-stiff ordinary differential equation (ODE) problems, thereby representing a significant advancement in the field of numerical analysis.