An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation

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

Chapter 1: Applied Mathematics and Mathematical Modeling

Chapter 2: Differential Equations

Part I: Conservation of Mass: Biology, Chemistry, Physics, and Engineering

Chapter 3: An Environmental Pollutant

Chapter 4: Acid Dissociation, Buffering, Titration, and Oscillation

Chapter 5: Reaction, Diffusion, and Convection

Chapter 6: Excitable Media: Transport of Electrical Signals on Neurons

Chapter 7: Splitting Methods

Chapter 8: Feedback Control

Chapter 9: Random Walks and Diffusion

Chapter 10: Problems and Projects: Concentration Gradients, Convection, Chemotaxis, Cruise Control, Constrained Control, Pearson’s Random Walk, Molecular Dynamics, Pattern Formation

Part II: Newton’s Second Law: Fluids and Elastic Solids

Chapter 11: Equations of Fluid Motion

Chapter 12: Flow in a Pipe

Chapter 13: Eulerian Flow

Chapter 14: Equations of Motion in Moving Coordinate Systems

Chapter 15: Water Waves

Chapter 16: Numerical Methods for Computational Fluid Dynamics

Chapter 17: Channel Flow

Chapter 18: Elasticity: Basic Theory and Equations of Motion

Chapter 19: 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

Part III: Electromagnetism: Maxwell’s Laws and Transmission Lines

Chapter 20: Classical Electromagnetism

Chapter 21: Transverse Electromagnetic (TEM) Mode

Chapter 22: Transmission Lines

Chapter 23: Problems and Projects: Waveguides, Lord Kelvin’s Model