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An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation introduces the reader to the methodology of modern applied mathematics in modeling, analysis, and scientific computing with emphasis on the use of ordinary and partial differential equations. Each topic is introduced with an attractive physical problem, where a mathematical model is constructed using physical and constitutive laws arising from the conservation of mass, conservation of momentum, or Maxwell's electrodynamics.
Relevant mathematical analysis (which might employ vector calculus, Fourier series, nonlinear ODEs, bifurcation theory, perturbation theory, potential theory, control theory, or probability theory) or scientific computing (which might include Newton's method, the method of lines, finite differences, finite elements, finite volumes, boundary elements, projection methods, smoothed particle hydrodynamics, or Lagrangian methods) is developed in context and used to make physically significant predictions. The target audience is advanced undergraduates (who have at least a working knowledge of vector calculus and linear ordinary differential equations) or beginning graduate students.
Readers will gain a solid and exciting introduction to modeling, mathematical analysis, and computation that provides the key ideas and skills needed to enter the wider world of modern applied mathematics.
Presents an integrated wealth of modeling, analysis, and numerical methods in one volumeProvides practical and comprehensible introductions to complex subjects, for example, conservation laws, CFD, SPH, BEM, and FEMIncludes a rich set of applications, with more appealing problems and projects suggested
Applied Mathematics and Mathematical Modeling
Differential Equations
An Environmental Pollutant
Acid Dissociation, Buffering, Titration, and Oscillation
Reaction, Diffusion, and Convection
Excitable Media: Transport of Electrical Signals on Neurons
Splitting Methods
Feedback Control
Random Walks and Diffusion
Problems and Projects: Concentration Gradients, Convection, Chemotaxis, Cruise Control, Constrained Control, Pearson’s Random Walk, Molecular Dynamics, Pattern Formation
Equations of Fluid Motion
Flow in a Pipe
Eulerian Flow
Equations of Motion in Moving Coordinate Systems
Water Waves
Numerical Methods for Computational Fluid Dynamics
Channel Flow
Elasticity: Basic Theory and Equations of Motion
Problems and Projects: Rods, Plates, Panel Flutter, Beams, Convection-Diffusion in Tunnels, Gravitational Potential of a Galaxy, Taylor Dispersion, Cavity Flow, Drag, Low and High Reynolds Number Flows, Free-Surface Flow, Channel Flow
Classical Electromagnetism
Transverse Electromagnetic (TEM) Mode
Transmission Lines
Problems and Projects: Waveguides, Lord Kelvin’s Model
Mathematical and Computational Notes
Answers to Selected Exercises