Addresses the construction, analysis, and intepretation of mathematical models that shed light on significant problems in the physical sciences. The authors' case studies approach leads to excitement in teaching realistic problems. The exercises reinforce, test and extend the reader's understanding. This reprint volume may be used as an upper level undergraduate or graduate textbook as well as a reference for researchers working on fluid mechanics, elasticity, perturbation methods, dimensional analysis, numerical analysis, continuum mechanics and differential equations.
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.
'The words 'applied mathematics' decorate the covers of many books these days, but precious little of the practice of applied mathematics seeps through into the pages bound between those covers. Of the select few that do indeed introduce the reader to applied mathematics, this book may well be the best. I know of none better.' Paul Davis, American Mathematical Monthly
