School of Mechanical Engineering and Safety Engineering

Summer Semester

Lecture: Numerical Mathematics

Lecturer: Prof. Dr.-Ing. Axel Schumacher

Lecture belongs to the module: Advanced mathematics

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Contents:

Basics for the application of important numerical methods of Engineering.

1. Introduction

1.1 Opportunities of Numeric?

1.2 Necesseties of Numeric

1.3 Project Preview: Explicit Time Integratioin

1.4 Project Preview: Numerical Optimization

2. Numerical Differentiation and Integration

2.1 Why Differentiation and Integration?

2.2 Scheme of Differentiation

2.3 Numerical Integration

3. Calculation of Non-Linear Equations

3.1 Interval Bisection Method

3.2 Newton-Method

4. Approximation-Method

4.1 Introduction

4.2 Basics of Errors and Regression

4.3 Calculation of the Root-Mean-Square Error Criterion

4.4 Polynom-Approximation

4.5 Taylor Series

4.6 Moderne Meta-Models

4.7 Geometrical Descriptioin

4.8 FOURIER-Analysis

5. Numerical Calculatioin for Differential Equations

5.1 Introduction

5.2 Explicit Euler-Method

5.3 Heun-Method

5.4 Runge-Kutta-Method

5.5 Comparisons

5.6 Calculation of First Order Differential Equations

5.7 Calculation of Second Order Differential Equations

5.8 Special Solutions of Coupled DLGs

5.9 Solution of Partial Differential Equations

6. Calculation of Systems of Linear Equations

6.1 Introduction

6.2 Generation of Systems of Equation

6.3 Gaussian Elimination

6.4 JACOBI-Method

6.5 Gauß-Seidel-Method

6.6 LR-Decomposition

6.7 Pivot Elements

6.8 Cholesky-Decomposition

6.9 General Guidelines for Solution Systems of Linear Equations

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