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Mathematics Applied to Deterministic Problems in the Natural Sciences

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By C.C. Lin and L.A. Segel


1988 / xxi + 609 pages / Softcover / ISBN: 978-0-898712-29-2 / List Price $73.50 / SIAM Member Price $51.45 / Order Code CL01

"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, February 1975 

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 many problems and 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.

Adopted for classroom use by Purdue University, New York University, Rensselaer Polytechnic Institute, and Claremont Graduate School.

Part 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 Partial 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 partial 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; Part 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; Part 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.

ISBN: 9780898712292

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