Analysis of Differentials Equations and Get their Solutions using Sobolev Space

Among many purposes of science, analyzing nature may be the most important and beautiful part. We all know that Differential Equations (DEs) are the mathematical expression of many natural phenomena. In modern science, analyzing tools like Calculus, Measure, sequence, series etc. have been used very frequently. Every analysis of DEs has only one goal which is to get the solutions of a DE. In fact, most of these DEs don’t have exact solutions and many methods has been introduced to get some good solutions. Now-a-days, Functional Analysis plays an important role to analyze these methods. Some methods have solid foundation and flexibility. To make use all of these methods properly, we have to understand the nature of DEs and also realize the characteristics of the solutions. Without having any idea about solutions we don’t think more, this is why we have started our study of analysis to take more and more benefits of these methods. This analysis will give us a solid platform to select best methods among others and will help us to find new more accurate methods. As usual, this analysis is covering the concepts of measure, which leads us to a more useful Integration concept. For finding the solutions of DEs some essential spaces have been introduced such as Lp spaces. The space concepts have been more specified into Sobolev Spaces.