The Application Of Regular Perturbation Theory In Mechanics

Regular perturbation is one of the most important methods to deal with the mathematics problem in the theory of perturbation. It is very important both for the research and application, because it has very rigorous and clear step and the request is always kept constant. The perturbation theory has been used in many research areas as a mathematic method, especially in the applied project. The theory of perturbation is an approximate analysis of the factor of disturbance and was originated from celestial mechanics. The basic idea of perturbation is to use the result of question B to explain an question A which is similar to question B, because the question A is harder than the question B. the theory of perturbation has been recognized as a important mathematics method because of its simply analysis steps.In this paper, we have shown the history and development of the theory of perturbation, further more the position and the future of that has been given too. In section 2, we have introduced the concept of perturbation problems, regular perturbation problem and scaling method, asymptoticexpansion,order symbols. The character and relative Computing Methods have also been shown. The general idea of the regular perturbation theory is:First, changing the problem to perturbation problems after get in the small parametersε; second, use the asymptoticexpansion method expanding the solution of equation, then we can make sure the situation of all items in the expansion; third, take the need small parametersεsubstituting in the expanding the solution of equation, so we can get the approximate solution of the original problem.Chapter 3 Consider the first solution of Riccati equation problem and Supposing the solution is y(x,ε)-y0 (x)+εy1(x)+ε2y2(x)+…. Consider the solution of general equation problem Supposing the solution is For example the first solution of nonlinear equation problem the small parametersε→0, asymptoticexpansion method expanding power series and and y can unlimited differential. As well as some practical applications of the mechanical problems:While the regular perturbation method is widely used, but on many issues is not very good application. Therefore, in Chapter 5 describes the regular perturbation method failure. Unlimited Domains: considered by the non-linear spring-mass vibration problem of constrained, Equation has singular points:Differential equation boundary condition y(1)= e-1.The type of change in a complex differential equation:The question of the Dirichlet functionφ(x,y,ε), its differential equation Boundary conditionEquation through specific practical problems, We can see that the regular perturbation method and its practical usefulness.