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.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Preface - p. xi 6. Acknowledgments - p. xiii 7. To the Professor - p. xv 8. To the Student - p. xix 9. Chapter 1: Applied Mathematics and Mathematical Modeling - p. xxi 10. Chapter 2: Differential Equations - p. 10 11. Part I: Conservation of Mass: Biology, Chemistry, Physics, and Engineering - p. 30 12. Chapter 3: An Environmental Pollutant - p. 30 13. Chapter 4: Acid Dissociation, Buffering, Titration, and Oscillation - p. 39 14. Chapter 5: Reaction, Diffusion, and Convection - p. 85 15. Abstract - p. 85 16. 5.1 Fundamental and constitutive model equations - p. 85 17. 5.2 Reaction-diffusion in one spatial dimension: heat, genetic mutations, and traveling waves - p. 88 18. 5.3 Reaction-diffusion systems: the gray–scott model and pattern formation - p. 113 19. 5.4 Analysis of reaction-diffusion models: qualitative and numerical methods - p. 116 20. 5.5 Beyond euler’s method for reaction-diffusion pde: diffusion of gas in a tunnel, gas in porous media, second-order in time methods, and unconditional stability - p. 151 21. Chapter 6: Excitable Media: Transport of Electrical Signals on Neurons - p. 195 22. Chapter 7: Splitting Methods - p. 221 23. Chapter 8: Feedback Control - p. 229 24. Chapter 9: Random Walks and Diffusion - p. 241 25. Chapter 10: Problems and Projects: Concentration Gradients, Convection, Chemotaxis, Cruise Control, Constrained Control, Pearson’s Random Walk, Molecular Dynamics, Pattern Formation - p. 277 26. Part II: Newton’s Second Law: Fluids and Elastic Solids - p. 291 27. Chapter 11: Equations of Fluid Motion - p. 293 28. Chapter 12: Flow in a Pipe - p. 321 29. Chapter 13: Eulerian Flow - p. 327 30. Chapter 14: Equations of Motion in Moving Coordinate Systems - p. 357 31. Chapter 15: Water Waves - p. 383 32. Chapter 16: Numerical Methods for Computational Fluid Dynamics - p. 403 33. Abstract - p. 403 34. 16.1 Approximations of incompressible navier–stokes flows - p. 403 35. 16.2 A numerical method for water waves - p. 446 36. 16.3 The boundary element method (BEM) - p. 450 37. 16.4 Boundary integral representation - p. 450 38. 16.5 Boundary integral equation - p. 455 39. 16.6 Discretization for BEM - p. 459 40. 16.7 Smoothed particle hydrodynamics - p. 475 41. 16.8 Simulation of a free-surface flow - p. 506 42. Chapter 17: Channel Flow - p. 511 43. Abstract - p. 511 44. 17.1 Conservation of mass - p. 513 45. 17.2 Momentum balance - p. 515 46. 17.3 Boundary layer theory - p. 527 47. 17.4 Flow in prismatic channels with rectangular cross sections of constant width - p. 535 48. 17.5 Hydraulic jump - p. 539 49. 17.6 Saint-venant model and systems of conservation laws - p. 542 50. 17.7 Surface waves - p. 564 51. Chapter 18: Elasticity: Basic Theory and Equations of Motion - p. 577 52. Abstract - p. 577 53. 18.1 The taut wire: separation of variables and fourier series for the wave equation - p. 589 54. 18.2 Longitudinal waves in a rod with varying cross section - p. 611 55. 18.3 Ultrasonics - p. 616 56. 18.4 A three-dimensional elastostatics problem: a copper block bolted to a steel plate - p. 620 57. 18.5 A one-dimensional elasticity model - p. 624 58. 18.6 Weak formulation of one-dimensional boundary value problems - p. 626 59. 18.7 One-dimensional finite element method discretization - p. 635 60. 18.8 Coding for the one-dimensional finite element method - p. 641 61. 18.9 Weak formulation and finite element method for linear elasticity - p. 647 62. 18.10 A three-dimensional finite element application - p. 653 63. 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 - p. 671 64. Part III: Electromagnetism: Maxwell’s Laws and Transmission Lines - p. 701 65. Chapter 20: Classical Electromagnetism - p. 703 66. Chapter 21: Transverse Electromagnetic (TEM) Mode - p. 719 67. Chapter 22: Transmission Lines - p. 731 68. Chapter 23: Problems and Projects: Waveguides, Lord Kelvin’s Model - p. 775 69. Mathematical and Computational Notes - p. 793 70. Answers to Selected Exercises - p. 827 71. References - p. 841 72. Index - p. 847
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.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Preface - p. xi 6. Acknowledgments - p. xiii 7. To the Professor - p. xv 8. To the Student - p. xix 9. Chapter 1: Applied Mathematics and Mathematical Modeling - p. xxi 10. Chapter 2: Differential Equations - p. 10 11. Part I: Conservation of Mass: Biology, Chemistry, Physics, and Engineering - p. 30 12. Chapter 3: An Environmental Pollutant - p. 30 13. Chapter 4: Acid Dissociation, Buffering, Titration, and Oscillation - p. 39 14. Chapter 5: Reaction, Diffusion, and Convection - p. 85 15. Abstract - p. 85 16. 5.1 Fundamental and constitutive model equations - p. 85 17. 5.2 Reaction-diffusion in one spatial dimension: heat, genetic mutations, and traveling waves - p. 88 18. 5.3 Reaction-diffusion systems: the gray–scott model and pattern formation - p. 113 19. 5.4 Analysis of reaction-diffusion models: qualitative and numerical methods - p. 116 20. 5.5 Beyond euler’s method for reaction-diffusion pde: diffusion of gas in a tunnel, gas in porous media, second-order in time methods, and unconditional stability - p. 151 21. Chapter 6: Excitable Media: Transport of Electrical Signals on Neurons - p. 195 22. Chapter 7: Splitting Methods - p. 221 23. Chapter 8: Feedback Control - p. 229 24. Chapter 9: Random Walks and Diffusion - p. 241 25. Chapter 10: Problems and Projects: Concentration Gradients, Convection, Chemotaxis, Cruise Control, Constrained Control, Pearson’s Random Walk, Molecular Dynamics, Pattern Formation - p. 277 26. Part II: Newton’s Second Law: Fluids and Elastic Solids - p. 291 27. Chapter 11: Equations of Fluid Motion - p. 293 28. Chapter 12: Flow in a Pipe - p. 321 29. Chapter 13: Eulerian Flow - p. 327 30. Chapter 14: Equations of Motion in Moving Coordinate Systems - p. 357 31. Chapter 15: Water Waves - p. 383 32. Chapter 16: Numerical Methods for Computational Fluid Dynamics - p. 403 33. Abstract - p. 403 34. 16.1 Approximations of incompressible navier–stokes flows - p. 403 35. 16.2 A numerical method for water waves - p. 446 36. 16.3 The boundary element method (BEM) - p. 450 37. 16.4 Boundary integral representation - p. 450 38. 16.5 Boundary integral equation - p. 455 39. 16.6 Discretization for BEM - p. 459 40. 16.7 Smoothed particle hydrodynamics - p. 475 41. 16.8 Simulation of a free-surface flow - p. 506 42. Chapter 17: Channel Flow - p. 511 43. Abstract - p. 511 44. 17.1 Conservation of mass - p. 513 45. 17.2 Momentum balance - p. 515 46. 17.3 Boundary layer theory - p. 527 47. 17.4 Flow in prismatic channels with rectangular cross sections of constant width - p. 535 48. 17.5 Hydraulic jump - p. 539 49. 17.6 Saint-venant model and systems of conservation laws - p. 542 50. 17.7 Surface waves - p. 564 51. Chapter 18: Elasticity: Basic Theory and Equations of Motion - p. 577 52. Abstract - p. 577 53. 18.1 The taut wire: separation of variables and fourier series for the wave equation - p. 589 54. 18.2 Longitudinal waves in a rod with varying cross section - p. 611 55. 18.3 Ultrasonics - p. 616 56. 18.4 A three-dimensional elastostatics problem: a copper block bolted to a steel plate - p. 620 57. 18.5 A one-dimensional elasticity model - p. 624 58. 18.6 Weak formulation of one-dimensional boundary value problems - p. 626 59. 18.7 One-dimensional finite element method discretization - p. 635 60. 18.8 Coding for the one-dimensional finite element method - p. 641 61. 18.9 Weak formulation and finite element method for linear elasticity - p. 647 62. 18.10 A three-dimensional finite element application - p. 653 63. 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 - p. 671 64. Part III: Electromagnetism: Maxwell’s Laws and Transmission Lines - p. 701 65. Chapter 20: Classical Electromagnetism - p. 703 66. Chapter 21: Transverse Electromagnetic (TEM) Mode - p. 719 67. Chapter 22: Transmission Lines - p. 731 68. Chapter 23: Problems and Projects: Waveguides, Lord Kelvin’s Model - p. 775 69. Mathematical and Computational Notes - p. 793 70. Answers to Selected Exercises - p. 827 71. References - p. 841 72. Index - p. 847
