Mathematics Applied to Deterministic Problems in the Natural Sciences

A: An Overview of the Interaction of Mathematics and Natural Science. Chapter 1: What is Applied Mathematics On the nature of applied mathematics Introduction to the analysis of galactic structure Aggregation of slime mold amebae Chapter 2: Deterministic Systems and Ordinary Differential Equations Planetary orbits Elements of perturbation theory, including Poincare's method for periodic orbits A system of ordinary differential equations Chapter 3: Random Processes and ial Differential Equations Random walk in one dimension Langevin's equation Asymptotic series, Laplace's method, gamma function, Stirling's formula A difference equation and its limit Further considerations pertinent to the relationship between probability and ial differential equations Chapter 4: Superposition, Heat Flow, and Fourier Analysis Conduction of heat Fourier's theorem On the nature of Fourier series Chapter 5: Further Developments in Fourier Analysis Other aspects of heat conduction Sturn Liouville systems Brief introduction to Fourier transform Generalized harmonic analysis B: Some Fundamental Procedures Illustrated on Ordinary Differential Equations. Chapter 6: Simplification, Dimensional Analysis, and Scaling The basic simplification procedure Dimensional analysis Scaling Chapter 7: Regular Perturbation Theory The series method applied to the simple pendulum Projectile problem solved by perturbation theory Chapter 8: Illustration of Techniques on a Physiological Flow Problem Physical formulation and dimensional analysis of a model for """"standing gradient"" osmotically driven flow A mathematical model and its dimensional analysis Obtaining the final scaled dimensionless form of the mathematical model Solution and interpretation Chapter 9: Introduction to Singular Perturbation Theory Roots of polynomial equations Boundary value problems for ordinary differential equations Chapter 10: Singular Perturbation Theory Applied to a Problem in Biochemical Kinetics Formulation of an initial value problem for a one enzyme one substrate chemical reaction Approximate solution by singular perturbation methods Chapter 11: Three Techniques Applied to the Simple Pendulum Stability of normal and inverted equilibrium of the pendulum A multiple scale expansion The phase plane C: Introduction to Theories of Continuous Fields. Chapter 12: Longitudinal Motion of a Bar Derivation of the governing equations One dimensional elastic wave propagation Discontinuous solutions Work, energy, and vibrations Chapter 13: The Continuous Medium The continuum model Kinematics of deformable media the material derivative The Jacobian and its material derivative Chapter 14: Field Equations of Continuum Mechanics Conservation of mass Balance of linear momentum Balance of angular momentum Energy and entropy On constitutive equations, covariance and the continuum model Chapter 15: Inviscid Fluid Flow Stress in motionless and inviscid fluids Stability of a stratified fluid Compression waves in gases Uniform flow past a circular cylinder Chapter 16: Potential Theory Equations of Laplace and Poisson Green's functions Diffraction of acoustic waves by a hole.