Abstract
The approach taken and the perturbation type both influence how differential equations are perturbed and what method is utilized to analyze the impact of the perturbation. This is true since the approach is impacted by each of these factors. The nonlinear change of parameters formula and the Lyapunov approach are frequently used [1]. The disturbances themselves must be compared to a benchmark, and if an ideal form of the disturbances already exists, it must be eliminated. These two tasks must be completed. This need needs to be satisfied before starting either of the two processes. Work is already underway on a number of projects that will be put into action [2, 3] in an effort to improve this terrible condition. A comparison theorem comparing the equations of unperturbation and perturbation systems are pertinent to the theory of perturbations is what this study aims to establish. The two approaches from the first section of this comparison were combined to produce a flexible method for maintaining the characteristics of disturbances. Our results demonstrate the inclusion of the Lyapunov function-based classical comparison theorem, which facilitates a more thorough investigation of perturbation theory. This will be accomplished by providing a specific example to demonstrate how our theorem aligns with our earlier theorem. To accomplish this, it will be demonstrated how our theory functions as a concrete illustration of our earlier theorem. To demonstrate how our theory works with the one we currently have, we will use a specific example. This will demonstrate how our theorem improves the precision of the one we already have. To demonstrate how this works, let's look at a fictitious scenario that illustrates the outcomes and the applicability of our theory.
Keywords— Lyapunov method, Non-linear variation of parameters, Perturbations, Conventional comparison theorem.
References
1. V. Lakshmikantham and S. Leela, differential and Integral inequalities, Theory end Applications, Vol.1, Academic Press, New York, 1969.
2. S. Bernfeld, G. Ladde and V. Lakshmikantham, On the classes of different systems with the desired behavior. Rend. Circ. Mat. Palermo, 21 (1972), 85- 97.
3. G. S. Ladde, Variational Comparison Theorem and Perturbations of Non-Linear Systems, Vol. 3, Academic Press, New York, 1999.