Method of Global Transformation and its Role in Turning Point Problems

In this monograph, we introduce an asymptotic technique which can be used to obtain uniform asymptotic solutions of certain types of second-order, linear ordinary differential equations which may arise in wave propagation problems in non-homogeneous media. Of particular interest are those physical phenomenon which lead to simple and non-simple turning point problems where the nature of the fundamental solution undergoes a sudden metamorphosis of character at the turning point. We begin by giving a few examples which under certain conditions admit turning point behavior. The general theory of the method of global transformation is then introduced. After giving a preview of the method, a historical account of the idea of comparison equation is given which systematically leads to our generalization of the concept. Next, we transform the original problem to an appropriate comparison equation using new dependent and independent variables. This leads to a non-linear differential equation in the transformed coordinate. This non-linear equation has the property that its solution can be determined in the form of uniformly convergent series. We take advantage of this unique property and develop an iterative technique to obtain the solution of the original problem. In applying our iterative technique, the zero -order solution so obtained reproduces exactly the dominant solutions as given by WKBJ or method of stationary phase. We then obtain the higher-order solutions of the non-linear equation which yield new transformations. Use of these transformations in conjunction with the comparison equation give us higher-order correction terms in the asymptotic series solution of the original problem. These higher -order terms correspond to the higher-order terms which one would obtain by using the method of steepest descent. After giving an account of the well-known connection problem using our zero-order solution of the Airy equation, we conclude by giving an illustrative example of application of our method to a simple turning point problem. We illustrate the simplicity of our technique by determining high-order correction terms in the asymptotic series of the far-field solution of the Airy equation.