Ramabhadra Vasudevan

Time Dependent Processes

In this chapter we will deal with time dependent transport of particles in a medium with given parametric values describing the interaction of the particles with the medium. Imbedding methods to study the emergent beams are dealt with in detail. Laplace transform techniques are also applied to arrive at the solutions both internal and external. When the velocity of the particles become large along with the collision cross section o in such a way that their ratio is finite, the transport problem in the limit reduces to the diffusion problem [1]. The study of the diffusion equation employing imbedding methods is presented briefly. The inversion of the Laplace transform is a subject which we do not discuss here, but refer the reader to elegant methods developed in Reference [2].

A useful approximation to ⁻2

Using differential approximation, we obtain a remarkably accurate representa- tion of e-9 as a sum of three exponentials.

Numerical Analysis: Quasilinearization and the estimation of differential operators from eigenvalues

Given a linear ordinary differential operator containing several unknown constants and a number of its eigenvalues, the values of the unknown constants are estimated. A precise formulation is provided, and an effective numerical procedure for solution is indicated. The results of some computational experiments are given.

Application to the Wave Equation

In this chapter we will take up the study of the wave equations in one dimension and study the propagation of the wave in a region with inhomogeneous properties of refractive index by analyzing the reflection and transmission functions for the region. In the previous chapter we studied these functions in the context of particle transport. Similar studies carry over in the case of wave propagation. An order of scattering analysis of the emergent and internal solutions leads to Bremmer series [1] solutions under certain conditions.

Eikonal Equation and the WKB Approximation

In this chapter we describe briefly the methods adopted to arrive at the solutions of the wave equation in one dimension by well known eikonal approximations [1], successive applications of the Liouville transformations [2], and the elegant matrix formalism developed in the monograph by Froman and Froman [3]. We will relate them with the successive approximations, arrived at by methods based on imbedding principles in later chapters.

Dynamic Programming and Solution of Wave Equations

In this chapter we use the dynamic programming techniques as applied to the various problems and arrive at the structure of the solutions of the second order equations without solving them. The variation diminishing properties of the Green’s functin, the unimodal nature of the solutions of the Sturm- Liouville equations are derived in Sections 1 and 2. In Section 3, variational equations for the characteristic functions and characteristic values are obtained, treating one of the limits of the interval of integration as the imbedding parameter.