Differential Equations

The aim of the Differential Equations (DE) course is to teach the basic tools for studying the behavior of solutions to differential equations. After a brief introduction containing examples of differential equations from physics, we will cover three main chapters: the existence-uniqueness of solutions and the calculation of exact solutions, the stability of solutions to a DDE, and methods of numerically solving DDEs. I) Exact solutions of differential equations 1. Definitions: Differential equations(DE), Solutions of a DE, Cauchy problems. 2. Existence-uniqueness of solutions. Grönwall's lemma. Cauchy-Lipschitz theorem. 3. Exact solutions for - scalar ODEs of order 1 with separable variables, - linear systems of homogeneous ODEs with constant coefficients, - linear scalar ODEs of order m, homogeneous with constant coefficients. 4. Methods for varying constants. Application to non-homogeneous 1st-order linear ODE systems. Wronskian. Application to scalar linear ODEs of order m. II) Stability of ODE solutions 1. Definition of Lyapunov stability, attractivity and asymptotic stability. 2. Study of the stability of the null solution of a linear ODE system. 3. Study the stability of a stationary solution of a nonlinear ODE by studying the linearized sound spectrum. 4. Lyapunov functions. Strict Lyapunov functions. III) Numerical methods 1. ODE discretization. Explicit and implicit Euler methods. Other one-step methods. 2. Definitions: Consistency, Order and Convergence. 3. Definitions: A-stability domain, unconditional A-stability for 1-step methods. 4. Explicit Runge-Kutta methods - Presentation of methods and calculation of order. - Stability function and A-stability domain of a Runge-Kutta method. 5. Newmark and Störmer-Verlet methods - Presentation of methods and calculation of order. - Conservation of angular momentum and long-time behavior.6. Linear multi-step methods - Adams-Bashforth, Adams-Moulton and BDF methods. - Order criteria for linear multi-step methods. - Dimension of the vector space of all numerical solutions obtained by a linear multistep method for a homogeneous linear ODE of order 1. Parasitic solutions. - 0-stability and A-stability of multi-step methods. - Dahlquist barriers.