Wave Propagation: An Invariant Imbedding Approach

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The SCQlldIII of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geometry interacts with . physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They· draw upon widely different sections of mathematics.

Table of Contents:
1 / Introduction.- 1. Introduction.- 2. Propagation of a Plane Electromagnetic Wave in a Stratified Medium.- 3. Basic Differential Equations of the Electromagnetic Field.- 4. Propagation of E.M. Waves through Multilayers.- 5. The Schrödinger Equation.- 6. The Rectangular Potential Barrier.- 7. The JWKB Solutions.- References.- II / Eikonal Equation and the WKB Approximation.- 1. Introduction.- 2. The Eikonal Expansion.- 3. Derivation of the Solution of the Schrödinger Equation using Matrix Methods.- 4. Asymptotic Behavior of the Solutions.- References.- III / Invariant Imbedding.- 1. Introduction.- 2. Invariant Imbedding Method.- 3. The Classical Approach.- 4. The Invariant Imbedding Approach for Particle Transport.- 5. Riccati Transformations.- 6. Linearization and Solution of the Riccati Equations.- 7. Conservation Relations.- 8. Scattering Matrix Formalism.- 9. Homogeneous Anisotropic Media Forming an Obstacle.- References.- IV / Application to the Wave Equation.- 1. Introduction.- 2. A Continuous Medium Problem.- 3. Bremmer Solutions.- 4. Coupled Differential and Integral Equations for the Two Beams.- 5. Convergence Properties of the Series Solutions.- 6. Bremmer Series Using Finite Order Scattering Reflection and Transmission Functions.- 7. Wave Equations with a Source Term.- References.- V / The Bremmer Series.- 1. Introduction.- 2. A New Type of Refractive Index Profile in Each Layer and the Reflection and Transmission Coefficients.- 3. Splitting of the Wave Function.- 4. Extensions to Other Types of Series.- References.- VI / Generalizations.- 1. Introduction.- 2. Method of Successive Diagonalization.- 3. Approximation to the Eikonal Solution Using Quasilinearization.- References.- VII / Time Dependent Processes.- 1. Introduction.- 2. Time Dependent TransportProblems.- 3. Transport Equation in the Limit of Large Velocities and Large ?.- 4. The Eigenvalue Problems.- 5. Eigenvalue Problems of Sturm-Liouville Systems.- 6. Time Dependent Wave Equation.- 7. Wiener Integrals.- References.- VIII / Asymptotic Properties.- 1. Introduction.- 2. Asymptotic Behavior of the Solutions of the Schrödinger Equation.- 3. The Phase Approach.- 4. Integral Equation Representation.- References.- IX / Operator Techniques.- 1. Introduction.- 2. The Baker-Campbell-Hausdorff Series.- 3. The Magnus Expansion.- 4. Higher Dimensional Wave Equations.- 5. Multidimensional Imbedding.- 6. Higher Order Equations.- References.- X / Variational Principles.- 1. Introduction.- 2. Bubnov-Galerkin Method.- 3. The Rayleigh-Ritz Method.- 4. Sturm-Liouville Theory.- 5. Rayleigh-Ritz Method and Physical Processes.- 6. The Maximum Functional.- 7. Dynamic Programming Method.- References.- XI / Dynamic Programming and Solution of Wave Equations.- 1. Introduction.- 2. Properties of the Green’s Function.- 3. The Sturm Oscillation Theorem and Unimodal Properties.- 4. Characteristic Values and Characteristic Functions.- 5. Determination of Characteristic Values of the Sturm-Liouville Equation.- 6. Another Type of Cauchy System for the Green’s Function and the Solution of Two Point Boundary Value Problem.- 7. Fredholm Resolvent.- 8. The Riccati Equation.- 9. Quasilinearization.- 10. The Cross-Ratio Relations.- 11. Matrix Riccati Equation and Auxiliary Functions.- References.- XII / Approximations.- 1. Introduction.- 2. Quadrature.- 3. Differential Quadrature.- 4. Determination of Weighting Coefficients.- 5. Higher Order Problems.- 6. Spline Approximation.- 7. Approximate Solutions.- 8. Segmental Curve Fitting.- 9. Dynamic Programming Approach.- 10. Splines Via DynamicProgramming.- 11. Derivation of Spline by Dynamic Programming.- 12. Equivalence of the Recursion Relations Obtained by Dynamic Programming and the Usual Results.- References.- Exercises and Notes.- Index of Names.- Index of Subjects.