This book will have strong appeal to interdisciplinary audiences, particularly in regard to its treatments of fluid mechanics, heat equations, and continuum mechanics. There is also a heavy focus on vector analysis. Maple examples, exercises, and an appendix is also included.
This book will have strong appeal to interdisciplinary audiences, particularly in regard to its treatments of fluid mechanics, heat equations, and continuum mechanics. There is also a heavy focus on vector analysis. Maple examples, exercises, and an appendix is also included.
1 Introduction.- 1.1 Vector Analysis: Some Basic Notions.- 1.2 General Systems of PDEs.- 1.3 The Main Examples of PDEs.- 1.4 The Flow Generated by a Vector Field.- 2 Derivation of the Heat Equation.- 2.1 Heat Flow: Fourier's Law.- 2.2 The Heat Equation (Without Convection).- 2.3 Initial Conditions and Boundary Conditions.- 2.4 Initial-Boundary Value Problems.- 3 The 1-D Heat Equation.- 3.1 The Homogeneous Problem.- 3.2 Separation of Variables.- 3.3 Sturm-Liouville Problems.- 4 Solution of the 1-D Heat Problem.- 4.1 Solution of the Homogeneous Heat IBVP.- 4.2 Transforming to Homogeneous BCs.- 4.3 The Semi-Homogeneous Heat Problem.- 5 Computational Analysis.- 5.1 Plotting Solutions of Heat Problems.- 5.2 Sturm-Liouville Problems.- 5.3 Fourier Analysis.- 6 Two-Dimensional Heat Flow.- 6.1 The 2-D Heat Equation.- 6.2 Rectangular Regions R.- 6.3 The Homogeneous Case.- 6.4 The Semi-Homogeneous Problem.- 6.5 Transforming to Homogeneous BCs.- 7 Boundary Value Problems.- 7.1 The Dirichlet Problem for the Unit Square.- 7.2 The Neumann Problem for the Unit Square.- 7.3 Mixed BVPs.- 8 3-D Heat Flow.- 8.1 The Heat Equation for the Unit Cube.- 8.2 The Semi-Homogeneous Problem.- 8.3 Transforming to Homogeneous BCs.- 8.4 Examples of BVPs for the Unit Cube.- 8.5 Heat Problems for the Cube.- 9 Maxwell's Equations.- 9.1 Maxwell's Equations in Empty Space.- 9.2 Electrostatics and Magnetostatics.- 9.3 Existence of Potentials.- 9.4 Potentials and Gauge Transformations.- 10 Fluid Mechanics.- 10.1 The Stress Tensor for a Fluid.- 10.2 The Fluid Equations.- 10.3 Special Solutions of the Euler Equations.- 10.4 Special Solutions of the Navier-Stokes Equations.- 10.5 Solution of the Navier-Stokes Equations.- 11 Waves in Elastic Materials.- 11.1 Strings: 1-D Elastic Materials.- 11.2 Membranes: 2-Dimensional Elastic Materials.- 11.3 Solids: 3-D Elastic Materials.- 12 The Heat IBVP in Polar Coordinates.- 12.1 The Polar Coordinate Map.- 12.2 The Laplacian in Polar Coordinates.- 12.3 The BCs and IC in Polar Coordinates.- 12.4 The Heat IBVP for a Disk.- 12.5 Solution of the Homogeneous Problem.- 12.6 The Heat Equation for an Annulus.- 12.7 The Homogeneous Problem for an Annulus.- 12.8 BVPs in Polar Coordinates.- 13 Solution of the Heat IBVP in General.- 13.1 Weak Solutions of Poisson's Equation.- 13.2 Solution of the Heat IBVP.- Appendix A Vector Analysis.- A.1 Curves and Line Integrals.- A.2 Surfaces and Surface Integrals.- A.3 Regions and Volume Integrals.- A.4 Boundaries.- A.5 Closed Curves and Surfaces.- A.6 The Theorems of Gauss, Green, and Stokes.- A.7 Curvatures and Fundamental Forms.- Appendix B Continuum Mechanics.- B.1 The Eulerian Description.- B.2 The Lagrangian Description.- B.3 Two-Dimensional Materials.- B.4 One-Dimensional Materials.- Appendix C Maple Reference Guide.- C.1 Mathematical Expressions and Functions.- C.2 Packages.- C.3 Plotting and Visualization.- C.4 Plotting Flows.- C.5 Programming.- Appendix D Symbols and Tables.- References.
1 Introduction.- 1.1 Vector Analysis: Some Basic Notions.- 1.2 General Systems of PDEs.- 1.3 The Main Examples of PDEs.- 1.4 The Flow Generated by a Vector Field.- 2 Derivation of the Heat Equation.- 2.1 Heat Flow: Fourier's Law.- 2.2 The Heat Equation (Without Convection).- 2.3 Initial Conditions and Boundary Conditions.- 2.4 Initial-Boundary Value Problems.- 3 The 1-D Heat Equation.- 3.1 The Homogeneous Problem.- 3.2 Separation of Variables.- 3.3 Sturm-Liouville Problems.- 4 Solution of the 1-D Heat Problem.- 4.1 Solution of the Homogeneous Heat IBVP.- 4.2 Transforming to Homogeneous BCs.- 4.3 The Semi-Homogeneous Heat Problem.- 5 Computational Analysis.- 5.1 Plotting Solutions of Heat Problems.- 5.2 Sturm-Liouville Problems.- 5.3 Fourier Analysis.- 6 Two-Dimensional Heat Flow.- 6.1 The 2-D Heat Equation.- 6.2 Rectangular Regions R.- 6.3 The Homogeneous Case.- 6.4 The Semi-Homogeneous Problem.- 6.5 Transforming to Homogeneous BCs.- 7 Boundary Value Problems.- 7.1 The Dirichlet Problem for the Unit Square.- 7.2 The Neumann Problem for the Unit Square.- 7.3 Mixed BVPs.- 8 3-D Heat Flow.- 8.1 The Heat Equation for the Unit Cube.- 8.2 The Semi-Homogeneous Problem.- 8.3 Transforming to Homogeneous BCs.- 8.4 Examples of BVPs for the Unit Cube.- 8.5 Heat Problems for the Cube.- 9 Maxwell's Equations.- 9.1 Maxwell's Equations in Empty Space.- 9.2 Electrostatics and Magnetostatics.- 9.3 Existence of Potentials.- 9.4 Potentials and Gauge Transformations.- 10 Fluid Mechanics.- 10.1 The Stress Tensor for a Fluid.- 10.2 The Fluid Equations.- 10.3 Special Solutions of the Euler Equations.- 10.4 Special Solutions of the Navier-Stokes Equations.- 10.5 Solution of the Navier-Stokes Equations.- 11 Waves in Elastic Materials.- 11.1 Strings: 1-D Elastic Materials.- 11.2 Membranes: 2-Dimensional Elastic Materials.- 11.3 Solids: 3-D Elastic Materials.- 12 The Heat IBVP in Polar Coordinates.- 12.1 The Polar Coordinate Map.- 12.2 The Laplacian in Polar Coordinates.- 12.3 The BCs and IC in Polar Coordinates.- 12.4 The Heat IBVP for a Disk.- 12.5 Solution of the Homogeneous Problem.- 12.6 The Heat Equation for an Annulus.- 12.7 The Homogeneous Problem for an Annulus.- 12.8 BVPs in Polar Coordinates.- 13 Solution of the Heat IBVP in General.- 13.1 Weak Solutions of Poisson's Equation.- 13.2 Solution of the Heat IBVP.- Appendix A Vector Analysis.- A.1 Curves and Line Integrals.- A.2 Surfaces and Surface Integrals.- A.3 Regions and Volume Integrals.- A.4 Boundaries.- A.5 Closed Curves and Surfaces.- A.6 The Theorems of Gauss, Green, and Stokes.- A.7 Curvatures and Fundamental Forms.- Appendix B Continuum Mechanics.- B.1 The Eulerian Description.- B.2 The Lagrangian Description.- B.3 Two-Dimensional Materials.- B.4 One-Dimensional Materials.- Appendix C Maple Reference Guide.- C.1 Mathematical Expressions and Functions.- C.2 Packages.- C.3 Plotting and Visualization.- C.4 Plotting Flows.- C.5 Programming.- Appendix D Symbols and Tables.- References.
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