Numerical Methods for Solving First-Order Nonlinear Differential Equations Using a Linear Block Approach

Abstract

This study presents a new block hybrid method aimed at solving nonlinear first-order initial value problems (IVPs). Nonlinear IVPs often appear in complex systems involving phenomena such as population dynamics, fluid flow and chemical reactions. These problems are characterized by chaotic behavior, multiple solutions and sensitivity to initial conditions, making analytical solutions difficult to obtain. To address this challenge, the study develops a block hybrid method using a linear block approach. The method employs polynomial interpolation to derive a continuous scheme, which is then discretized to generate the block method. The method is analyzed to determine its properties, including consistency, zero-stability and convergence. The block method is found to have an order of seven and the corresponding error constant is computed to demonstrate its high accuracy. A stability function is derived to examine the method's behavior and the region of absolute stability is plotted to verify its performance for stiff equations. Numerical experiments are conducted to validate the method, with results compared to existing techniques. The new block hybrid method exhibits superior accuracy, efficiency and stability when applied to nonlinear first-order IVPs, outperforming traditional methods in terms of reduced error and computational effort. This method offers a valuable alternative for solving complex nonlinear differential equations.