Stable Computer Method for Solving Initial Value Problems with Engineering Applications

Abstract

Engineering and applied mathematics disciplines that involve differential equations in general, and initial value problems in particular, include classical mechanics, thermodynamics, electromagnetism, and the general theory of relativity. A reliable, stable, efficient, and consistent numerical scheme is frequently required for modelling and simulation of a wide range of real-world problems using differential equations. In this study, the tangent slope is assumed to be the contra-harmonic mean, in which the arithmetic mean is used as a correction instead of Euler's method to improve the efficiency of the improved Euler's technique for solving ordinary differential equations with initial conditions. The stability, consistency, and efficiency of the system were evaluated, and the conclusions were supported by the presentation of numerical test applications in engineering. According to the stability analysis, the proposed method has a wider stability region than other well-known methods that are currently used in the literature for solving initial-value problems. To validate the rate convergence of the numerical technique, a few initial value problems of both scalar and vector valued types were examined. The proposed method, modified Euler explicit method, and other methods known in the literature have all been used to calculate the absolute maximum error, absolute error at the last grid point of the integration interval under consideration, and computational time in seconds to test the performance. The Lorentz system was used as an example to illustrate the validity of the solution provided by the newly developed method. The method is determined to be more reliable than the commonly existing methods with the same order of convergence, as mentioned in the literature for numerical calculations and visualization of the results produced by all the methods discussed, Mat Lab-R2011b has been used.