A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

Abstract

Linear and nonlinear numerical techniques are the most popular techniques for finding approximate solutions to initial value problems in numerous scientific fields. Due to the substantial importance of ordinary differential equations, an attempt has been made in the present research study to obtain a new nonlinear hybrid technique based upon contraharmonic and harmonic means having fourth-order accuracy. Theoretical analysis in terms of consistency, stability, asymptotic errors (local and global truncation errors), and convergence has also been carried out. The newly formulated technique is compared with some existing techniques having the same characteristics and observed to be much better because of errors, CPU time, and stability region. The adaptive step-size approach improves the performance of the proposed technique, and strategies to control the errors are developed. Some numerical experiments for scalar and vector initial value problems, including logistic growth, sinusoidal and industrial Robot Arm systems, are presented to show better performance of the proposed technique.