## 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.

• 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.