Numerical Methods for Differential Equations

Code no: (3293) Semester: 6th, Teaching hours: 4

• Introduction
• Initial value Problems for ordinary differential equations. Single step methods (Euler, Taylor, Runge-
Kutta) and multi-step methods: construction, convergence, error estimates, stability. Systems of
differential equations.
• Examples of application of the Galerkin method with finite elements in one-dimensional two-point
boundary value problems and two-dimensional Dirichlet problems.
• Hilbert spaces, Projection theorem.
• Boundary value problems and Galerkin method: General weak form. The Lax-Milgram theorem. The
Galerkin theorem. Error estimates. Rayleigh-Ritz-Galerkin method.
• Weak derivatives and Sobolev spaces. Green formulas.
• Elliptic boundary problems. Existence and uniqueness. Mixed boundary conditions.
• Finite element methods for elliptic boundary value problems: One-dimensional finite elements.
Piecewise polynomial functions. Hermite cubic functions and splines. Two and three-dimensional
finite elements. Piecewise polynomial functions. Error estimates.
• Applications for electrical engineers.
• Finite element methods for evolutionary boundary value problems: Parabolic and Hyperbolic
Problems.
• Finite differences: One-dimensional two-point boundary value problems, Non-homogeneous
Dirichlet problem for Poisson equation, Heat equation problem (FTCS and Crank-Nicolson methods,
Stability), hyperbolic ploblems.