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AM351: ODE II Spring 2008

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2 AM 351 ODE II
Contents 1 Theory of higher order linear ODEs 5 1.1 Existence and uniqueness of solutions of nth order linear ODEs 5 1.2 Homogeneous ODEs and superposition theorem . . . . . . . . 6 1.2.1 Linear operator . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 The superposition theorem . . . . . . . . . . . . . . . . 7 1.3 Solutions of inhomogeneous ODES . . . . . . . . . . . . . . . 11 1.4 Qualitative properties of solutions . . . . . . . . . . . . . . . . 12 1.4.1 Equilibrium solutions . . . . . . . . . . . . . . . . . . . 12 1.4.2 Oscillatory solutions . . . . . . . . . . . . . . . . . . . 14 2 Theory of first order linear systems of ODEs 21 2.1 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Reformulating higher order ODEs . . . . . . . . . . . . 21 2.1.2 Existence and Uniqueness . . . . . . . . . . . . . . . . 22 2.1.3 General solutions of the homogeneous problem . . . . . 22 2.2 General solutions of the inhomogeneous problem . . . . . . . . 24 2.3 How to solve homogeneous first order linear systems of ODEs with constant coefficients . . . . . . . . . . . . . . . . . . . . . 25 2.4 Fundamental matrices and the solution to the inhomogeneous ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 The matrix exponential . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Fundamental matrices and exponential matrices . . . . 28 2.5.2 Calculating the matrix exponential . . . . . . . . . . . 30 2.6 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . 34 3 Existence and Uniqueness. Well posed problems 35 3.1 Local and Global existence theorems . . . . . . . . . . . . . . 35 3.1.1 Vector integral equation . . . . . . . . . . . . . . . . . 35 3

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4 AM 351 ODE II 3.1.2 Picard iteration (or successive approximation) . . . . . 36 3.1.3 Global and local existence theorems . . . . . . . . . . . 36 3.2 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Power Series 41 5 Numerical approximation of IVPs 43 5.1 Basics of numerical approximation . . . . . . . . . . . . . . . . 43 5.1.1 The Forward Euler Method (FE) . . . . . . . . . . . . 43 5.1.2 The Backward Euler Method (BE) . . . . . . . . . . . 44 5.1.3 The Midpoint Rule (MR) . . . . . . . . . . . . . . . . 45 5.2 Convergence of numerical schemes . . . . . . . . . . . . . . . . 45 5.3 Linear Multistep Methods (LM) . . . . . . . . . . . . . . . . . 47 5.3.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3.3 Convergence of LM schemes: Dahlquist theorem. . . . 50 5.3.4 Long-time stability . . . . . . . . . . . . . . . . . . . . 51 5.4 Non Linear One step Methods (NLO) . . . . . . . . . . . . . . 52 5.4.1 Definition of NLO . . . . . . . . . . . . . . . . . . . . . 52 5.4.2 Convergence of NLO methods . . . . . . . . . . . . . . 53 5.4.3 Local and global order of NLO schemes . . . . . . . . . 54 5.4.4 Long-time stability of NLO schemes . . . . . . . . . . . 54 5.5 Note on the necessity of long-time stability . . . . . . . . . . . 55 6 Introduction to Perturbation Theory 57 6.1 Regular perturbation method . . . . . . . . . . . . . . . . . . 57 6.2 Limitations of the regular perturbation method . . . . . . . . 59 7 Boundary Value Problems 65 7.1 Linear two-points BVPs . . . . . . . . . . . . . . . . . . . . . 65 7.1.1 Existence and uniqueness result . . . . . . . . . . . . . 65 7.1.2 The Fredholm alternative theorem . . . . . . . . . . . . 65 7.1.3 The shooting method . . . . . . . . . . . . . . . . . . . 67
Chapter 1 Theory of higher order linear ODEs In this chapter we will concentrate on linear ODEs of the form : P n ( x ) y ( n ) ( x ) + P n - 1 ( x ) y ( n - 1) ( x ) + ... + P 0 ( x ) y ( x ) = F ( x ) (1.1) 1.1 Existence and uniqueness of solutions of nth order linear ODEs Definition 1.1 A solution on the interval I of R of (1.1) is an n time differentiable function whose derivatives exist and satisfy (1.1) x ∈ I . Definition 1.2 The general solution on the interval I of R of (1.1) is a function Φ( x ; c 1 , c 2 , ..., c n ) involving n arbitrary constants, which represents all non singular solutions of (1.1) on I . Remark 1.1 Those n constants represent for instance the n initial condi- tions for an initial value problem. Example: see class notes.

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