**2nd module**

**Prof. Vincenzo Casulli**

**A.A. 1998/99**

**PROGRAMME**

5. Boundary Value Problems for Ordinary Differential Equations: Central Difference Method for Linear Boundary Value Problems, Upwind Difference Method for Linear Boundary Value Problems, Numerical Solution of Mildly Nonlinear Boundary Value Problems, Convergence of Difference Methods for Boundary Value Problems, The Finite Element Method, Differential Eigenvalue Problems.

6. Elliptic Equations: Boundary Value Problems for the Laplace Equation, Approximate Solution of the Dirichlet Problem on a Rectangle, Approximate Solution of the Dirichlet Problem on a General Domain, The General Linear Elliptic Equation, Upwind Difference Method for General Linear Elliptic Equations, Theory for the Numerical Solution of Linear Boundary value Problems, Numerical Solution of Mildly Nonlinear Problems.

7. Parabolic Equations: An Explicit Numerical Method for the Heat Equation, The General Linear Parabolic Equation, An Explicit Upwind Method, Numerical Solution of Mildly Nonlinear Problems, Convergence of Explicit Finite Difference Methods, Implicit Central Difference Method, Implicit Upwind Method, The Crank-Nicolson Method.

8. Hyperbolic Equations: The Cauchy Problem, An Explicit Method for Initial-Boundary Problems, An Implicit Method for Initial-Boundary Problems, Mildly Nonlinear Problems, Hyperbolic Systems, The Method of Characteristics for Initial Value Problems, The Method of Courant, Isaacson and Rees, The Lax-Wendroff Method.

**REFERENCE TEXTS**

D.GREENSPAN, V.CASULLI, Numerical Analysis for Applied Mathematics,
Science and Engineering, Addison-Wesley, 1988