Modeling and Simulation - Course Exercises

This repository contains the complete set of exercises and solutions for the Modeling and Simulation course conducted during the Winter Semester 2019/20 at the University of Paderborn as part of the M.Sc. program.

Course Overview

This course covers fundamental and advanced topics in numerical modeling and simulation, progressing from basic MATLAB programming to sophisticated numerical methods for solving differential equations and optimization problems.

Exercise Topics

Core Exercises

Exercise 1: Matlab Fundamentals

• MATLAB programming basics
• Matrix operations and functions
• Diagnostic tools and debugging

Exercise 2: Matlab Fundamentals, Modeling on Real Physical Problems

• Array manipulation and coordinate transformations
• Spherical to Cartesian coordinate conversions
• Real-world physical problem modeling

Exercise 3: Representation of Numbers in Digital Computers

• Binary number systems
• Number representation and conversion algorithms
• Digital precision and accuracy

Exercise 4: Errors, Approximations, and Taylor Series

• Numerical error analysis
• Taylor series approximations
• Floating-point arithmetic considerations

Exercise 5: ODE & Modeling and Simulation of Projectile's Motion in Air

• Ordinary differential equations for projectile motion
• Air resistance and drag forces
• Trajectory simulation and analysis

Exercise 6: Solution of Ordinary Differential Equations using Euler Methods

• Euler method fundamentals
• Forward and backward Euler methods
• Stability analysis and error estimation

Exercise 7: Numerical Methods for Solution of ODE

• Advanced numerical integration techniques
• Runge-Kutta methods introduction
• Comparative analysis of ODE solvers

Exercise 8: Ordinary Differential Equations and A-Stability

• Stability analysis of numerical methods
• A-stability criteria and stiff equations
• Implicit methods for stable solutions

Exercise 9: Shooting Method for ODEs

• Boundary value problems
• Shooting method implementation
• Newton-Raphson method for root finding

Exercise 9a: Finite Difference for ODEs

• Finite difference approximations
• Discretization of differential equations
• Boundary condition handling

Exercise 10: 4th Order RK Method for Solution of ODEs and Optimization Problems

• Classical Runge-Kutta method (4th order)
• Predator-prey population dynamics
• Optimization techniques and applications

Exercise 11: Forward Time Centered Space Method for 2D Heat Equation

• Partial differential equations
• FTCS method for heat diffusion
• 2D grid-based simulations

Exercise 12: Differential Equations & Finite Difference Time Domain Method

• FDTD method fundamentals
• Advanced PDE solution techniques
• Time-domain electromagnetic simulations

Exercise 13: Discrete Models

• Graph theory applications
• Dijkstra's shortest path algorithm
• Network optimization and routing

Topic Relationships and Progression

The exercises follow a logical progression building upon fundamental concepts:

Ex1-2 (MATLAB Basics) → Ex3 (Number Systems) → Ex4 (Numerical Errors)
                                   ↓
                           Ex5-6 (ODE Fundamentals) → Ex7-8 (Advanced ODE Methods)
                                   ↓
                           Ex9-9a (Boundary Value Problems) → Ex10 (RK Methods)
                                   ↓
                           Ex11-12 (PDE Methods) → Ex13 (Discrete Models)

Technical Implementation

The exercises demonstrate various numerical techniques using:

• MATLAB implementations for numerical algorithms
• Real-world physical problem modeling
• Comparative analysis of different methods
• Visualization and simulation results

Course Context

• Institution: University of Paderborn
• Program: M.Sc. (Master of Science)
• Semester: Winter Semester 2019/20 (October - March)
• Duration: 6 months of progressive learning

File Structure

Each exercise directory contains:

• Problem statement PDFs (ignored by gitignore)
• MATLAB script solutions and implementations
• Sample code and template files
• Visualization results and plots
• Complete working solutions

Prerequisites

Basic understanding of:

• Calculus and differential equations
• Linear algebra
• Numerical analysis concepts
• Programming fundamentals (MATLAB)

Learning Outcomes

Upon completion, students will have practical experience with:

• Numerical methods for ODEs and PDEs
• Stability analysis of numerical algorithms
• Optimization techniques and applications
• Discrete mathematics and graph algorithms
• Real-world problem simulation and modeling