Time-dependent Partial Differential Equations and Their Numerical Solution

1 Cauchy Problems.- 1.1 Introductory Examples.- 1.2 Well-Posedness.- 1.3 Hyperbolic Systems with Constant Coefficients.- 1.3.1 In One Space Dimension.- 1.3.2 Symmetrizer.- 1.3.3 Multiple Space Dimensions.- 1.4 General Systems with Constant Coefficients.- 1.4.1 Parabolic Systems.- 1.4.2 Mixed Systems.- 1.5 Linear Systems with Variable Coefficients.- 1.6 Remarks.- 2 Half Plane Problems.- 2.1 Hyperbolic Systems in One Dimension.- 2.2 Hyperbolic Systems in Two Dimensions.- 2.3 Well-Posed Half Plane Problems.- 2.4 Well-Posed Problems in the Generalized Sense.- 2.5 Farfield Boundary Conditions.- 2.6 Energy Estimates.- 2.7 First Order Systems with Variable Coefficients.- 2.8 Remarks.- 3 Difference Methods.- 3.1 Periodic Problems.- 3.2 Half Plane Problems.- 3.2.1 Energy Estimates.- 3.2.2 Estimates by using Laplace Transform.- 3.2.3 Error Estimates.- 3.3 Method of Lines.- 3.4 Remarks.- 4 Nonlinear Problems.- 4.1 General Discussion.- 4.2 Initial Value Problems for Ordinary Differential Equations.- 4.3 Existence Theorems for Nonlinear Partial Differential Equations.- 4.4 Perturbation Expansion.- 4.5 Convergence of Difference Methods.- 4.6 Remarks.