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# Student Group

PublicÂ·11 members
Jackson Brown

The appearance of ordinary and partial differential equations plays a significant part in various fields of sciences, practical physics, chemistry, mathematics, and biology. Qualitatively, there is physical relevance of the situation that determines the dynamical behaviors. Population expansion, potential fields, electric circuits, tree biological nature, and so on are all examples of the uses of practical physics. Differential equations are derived from physical situations. The linear differential equation solution is relatively simple, but finding an analytical analysis of a nonlinear differential equation could be difficult in many situations. Therefore, because most differential equations do not have an exact closed form solution, approximation and numerical approaches are regularly employed. Temporarily, many non-linear equations do not have a small parameter, but any traditional perturbation technique requires it. Therefore, this difficulty constricts the use of these perturbation techniques. A small parameter determination is a difficult procedure that requires an implementation of special procedures. Aimed at explaining ordinary nonlinear differential equations, the semi-analytical HPM can be a useful tool. He11 was the first Mathematician who proposed this method to solve nonlinear differential equations. The HPM has all the benefits of the perturbation approach without any necessity for a small parametric hypothesis. This approach overcomes calculation complexity, requires less computer memory, and has a faster calculation time than the previous methods. Accordingly, it is simple, powerful, effective and promising. The method requires initial conditions and generates an indefinite numerical as an analytical approximation. The HPM has been employed to analyze nonlinear differential equations in a number of investigations. The HPM was adapted by El-Dib and Moatimid12 to find accurate solutions for various forms in linear and nonlinear differential equations. The primary idea in their approach is coming up with an appropriate trial function, which is commonly expressed in terms of a power series. The cancellation of the first-order approximation solution ensures that all the advanced levels are likewise ignored. Consequently, the accuracy of the fixed zero-order solution will be confirmed the exact solution. The HPM was utilized by Moatimid13 to get an analytical approximate solution for a sliding bead in a smooth parabola. Due to the motivation for analyzing the Duffing oscillator on a variety of physics and engineering processes, the stability of a Duffing oscillator was analyzed by Moatimid14. It should be noticed that the present problem differs from those revealed by Moatimid13,14 in the structure of the model and well as the stability analysis. Additionally, the presented perturbed solution has been verified by RK4, and this has not done previously. Using the HPM, the principal equation of motion, the stability analysis, and many analytical approximate solutions were developed. The same method is utilized by Amer et al.15, and He et al.16 to obtain the desired approximate solutions. Tian and Wang17 developed a stability problem of linear time-delay system. Firstly, they described a generalized vector multiple integral inequality that can interpret several results as exceptional circumstances. Secondly, a delay-dependent stability (DDS) criterion for time-delay systems was constructed using these multiples. The DDS problem for a time-varying delay linear system was proposed by Tian and Wang18. They demonstrated that their method is more practical for dealing with time-varying delay systems. The authors provided a numerical example to demonstrate the utility of the stability criterion.