Mathematics Applied to Deterministic Problems in the Natural Sciences

'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

• A: An Overview of the Interaction of Mathematics and Natural Science. Chapter 1: What is Applied MathematicsOn the nature of applied mathematicsIntroduction to the analysis of galactic structureAggregation of slime mold amebaeChapter 2: Deterministic Systems and Ordinary Differential EquationsPlanetary orbitsElements of perturbation theory, including Poincare's method for periodic orbitsA system of ordinary differential equationsChapter 3: Random Processes and ial Differential EquationsRandom walk in one dimensionLangevin's equationAsymptotic series, Laplace's method, gamma function, Stirling's formulaA difference equation and its limitFurther considerations pertinent to the relationship between probability and ial differential equationsChapter 4: Superposition, Heat Flow, and Fourier AnalysisConduction of heatFourier's theoremOn the nature of Fourier seriesChapter 5: Further Developments in Fourier AnalysisOther aspects of heat conductionSturn Liouville systemsBrief introduction to Fourier transformGeneralized harmonic analysisB: Some Fundamental Procedures Illustrated on Ordinary Differential Equations. Chapter 6: Simplification, Dimensional Analysis, and ScalingThe basic simplification procedureDimensional analysisScalingChapter 7: Regular Perturbation TheoryThe series method applied to the simple pendulumProjectile problem solved by perturbation theoryChapter 8: Illustration of Techniques on a Physiological Flow ProblemPhysical formulation and dimensional analysis of a model for """"standing gradient"" osmotically driven flowA mathematical model and its dimensional analysisObtaining the final scaled dimensionless form of the mathematical modelSolution and interpretationChapter 9: Introduction to Singular Perturbation TheoryRoots of polynomial equationsBoundary value problems for ordinary differential equationsChapter 10: Singular Perturbation Theory Applied to a Problem in Biochemical KineticsFormulation of an initial value problem for a one enzyme one substrate chemical reactionApproximate solution by singular perturbation methodsChapter 11: Three Techniques Applied to the Simple PendulumStability of normal and inverted equilibrium of the pendulumA multiple scale expansionThe phase planeC: Introduction to Theories of Continuous Fields. Chapter 12: Longitudinal Motion of a BarDerivation of the governing equationsOne dimensional elastic wave propagationDiscontinuous solutionsWork, energy, and vibrationsChapter 13: The Continuous MediumThe continuum modelKinematics of deformable mediathe material derivativeThe Jacobian and its material derivativeChapter 14: Field Equations of Continuum MechanicsConservation of massBalance of linear momentumBalance of angular momentumEnergy and entropyOn constitutive equations, covarianceand the continuum modelChapter 15: Inviscid Fluid FlowStress in motionless and inviscid fluidsStability of a stratified fluidCompression waves in gasesUniform flow past a circular cylinderChapter 16: Potential TheoryEquations of Laplace and PoissonGreen's functionsDiffraction of acoustic waves by a hole.