Partial Differential Equations for Computational Science

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.