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# Ordinary and Partial Differential Equations pdf |physicscastle

## Ordinary and Partial Differential Equations pdf free

John W. Cain expresses profound gratitude to his advisor, Dr. David G. Schaeffer, James B. Duke Professor of Mathematics at Duke University, The first five chapters are based in part upon Professor Schaeffer’s introductory graduate course on ordinary differential equations The material has been adapted to accommodate upper-level undergraduate students, essentially by omitting technical proofs of the major theorems and including additional examples. Other major influences on this book include the excellent texts of Perko , Strauss , and Strogatz . In particular, the material presented in the last five chapters (including the ordering of the topics) is based heavily on Strauss’ book. On the other hand, our exposition, examples, and exercises are more “user-friendly”, making our text more accessible to readers with less background in mathematics, Dr. Reynolds dedicates her portion of this textbook to her mother, father and sisters, she thanks them for all their support and love, Finally Dr. Cain dedicates his portion of this textbook to his parents Jeanette and Harry, who he loves more than words can express.

### Contents of Partial Differential Equation

1 Introduction
1.1 Initial and Boundary Value Problems
2 Linear, Constant-Coefficient Systems
2.1 Homogeneous Systems
2.1.1 Diagonalizable Matrices
2.1.2 Algebraic and Geometric Multiplicities of Eigenvalues
2.1.3 Complex Eigenvalues
2.1.4 Repeated Eigenvalues and Non-Diagonalizable Matrices
2.2 Phase Portraits and Planar Systems
2.3 Stable, Unstable, and Center Subspaces
2.4 Trace and Determinant .
2.5 Inhomogeneous Systems
3 Nonlinear Systems: Local Theory
3.1 Linear Approximations of Functions of Several Variables
3.2 Fundamental Existence and Uniqueness Theorem
3.3 Global Existence, Dependence on Initial Conditions
3.4 Equilibria and Linearization
3.5 The Hartman-Grobman Theorem
3.6 The Stable Manifold Theorem
3.7 Non-Hyperbolic Equilibria and Lyapunov Functions
4 Periodic, Heteroclinic, and Homoclinic Orbits
4.1 Periodic Orbits and the Poincaré-Bendixon Theorem
4.2 Heteroclinic and Homoclinic Orbits
5 Bifurcations 140
5.1 Three Basic Bifurcations
5.2 Dependence of Solutions on Parameters
5.3 Andronov-Hopf Bifurcations
6 Introduction to Delay Differential Equations
6.1 Initial Value Problems
6.2 Solving Constant-Coefficient Delay Differential Equations
6.3 Characteristic Equations
6.4 The Hutchinson-Wright Equation
7 Introduction to Difference Equations
7.1 Basic Notions
7.2 Linear, Constant-Coefficient Difference Equations
7.3 First-Order Nonlinear Equations and Stability
7.4 Systems of Nonlinear Equations and Stability
7.5 Period-Doubling Bifurcations
7.6 Chaos
7.7 How to Control Chaos
8 Introduction to Partial Differential Equations
8.1 Basic Classification of Partial Differential Equations
8.2 Solutions of Partial Differential Equations
8.3 Initial Conditions and Boundary Conditions
8.4 Visualizing Solutions of Partial Differential Equations
9 Linear, First-Order Partial Differential Equations 236
9.1 Derivation and Solution of the Transport Equation
9.2 Method of Characteristics: More Examples
10 The Heat and Wave Equations on an Unbounded Domain
10.1 Derivation of the Heat and Wave Equations
10.2 Cauchy Problem for the Wave Equation
10.3 Cauchy Problem for the Heat Equation
10.4 Well-Posedness and the Heat Equation
10.5 Inhomogeneous Equations and Duhamel’s Principle