Computeraidedsolutionsinasymptoticsand Perturbationmethods

Asymptotics and perturbation theory are used widely in mathematics, physics, and other branches of science and technology, it is a very powerful tool for dealing with problems by obtaining approximate solutions of systems with some parameters very large or small. Major progresses were made in 21th century in the study of celestial mechanics and fluid mechanics. One of the most important technique is singular perturbation methods. In most cases not only does it give local results for the problem but also shows the global behavior of the solution.Numerical methods have been applied to solved many problems, but its applications are limited in the real world problems for the reason that the real problems in science and technology are usually containing many free parameters, and these parameters are varying within wide domains, but numerical methods could only deal with finite ranges and misuse numerical methods could give nonsensical solutions to real world problems. Thus, we hardly see any practical usage of numerical methods in theoretical mathematics, physics and other sciences. Even in the applied sciences, to decide what is"large"and"small"is a problem of experience and intuition, but experience and intuition are parts of science and they are unreliable. Instead, mathematicians and physicists apply Asymptotics and perturbation methods in their theoretical research, while applied mathematicians, scientists and engineers use Asymptotics and perturbation methods together with numerical methods and computers to solve their problems. In this work we introduce computer technique to asymptotics and perturbation theory free of the limitations associated with numerical methods, and we present Maple, with its easy availability, easy to use, powerful symbolic, graphic and animation capabilities, is one of the three most popular software packages in mathematics. This paper begins with a brief introduction of several methods of asymptotic analysis and perturbation theory, then we demonstrate that it is possible to use Maple to carry out these large scale calculations symbolically. This work intends to be as a model to show how to do mathematics in the computer age.