Mathematical Modelling in Engineering

Data is displayed for the academic year: 2025./2026.

Lecturers

Assoc. Prof.
Dario Bojanjac
PhD
Assoc. Prof.
Mario Bukal
PhD

Course Description

In order to better understand and predict future phenomena, the real world is described using mathematical models. A mathematical model is a description of a system or phenomenon using mathematical language, and the process of developing a mathematical model is called mathematical modeling. Since the real world is too complex to model its parts or phenomena in their original form, when developing a mathematical model, we choose which properties of the real world to consider and which to ignore. A mathematical model is an approximation of the real world, which typically does not describe real-world phenomena with complete precision, but with a good model, the solutions it provides for the purpose for which it was developed will be sufficiently close to the real world. The basis of mathematical modeling is identifying the causal relationships between input parameters and the phenomenon being modeled. Using mathematical language to record the selected causal relationships and assumptions upon which the model is based gives us access to a range of mathematical theories and techniques, as well as computational methods for solving the problem. The goal of the course Mathematical Modeling in Engineering is to familiarize students with basic techniques of mathematical modeling, such as dimensional analysis, applied asymptotic analysis, and the perturbation method, and their application to specific physical and engineering problems. In the course, students will learn how to answer questions such as: How does an epidemic spread? What is herd immunity? Why does fishing help preserve fish populations? How does heat spread? What power does a laser need to successfully drill through metal? How can we filter contaminated water? And many others.

Prerequisites

Understanding of basic concepts from Mathematical Analysis 1, primarily derivatives and integrals, as well as the application of derivatives in calculating the tangent to a curve."

Study Programmes

University undergraduate
[FER3-HR] Computing - study
Free Elective Courses (5. semester)
[FER3-HR] Electrical Engineering and Information Technology - study
Free Elective Courses (5. semester)

Learning Outcomes

  1. Define the concept of a mathematical model.
  2. Identify the main influences and interactions in the process we are modeling.
  3. Describe a process using a mathematical model.
  4. Solve and analyze the solutions of the mathematical model.
  5. Compare different methods of analyzing and solving the mathematical model.

Forms of Teaching

Lectures

Two hours of lectures are held weekly, during which the principles of mathematical modeling are addressed using a 'case study' approach. Up to an additional 10% of the course grade can be earned through active participation in class.

Exercises

One hour of tutorial sessions is held weekly, during which the methods and techniques developed in the lectures are further explored.

Consultations

Each student will be able to choose whether to earn up to a maximum of 30% of the grade by solving homework assignments or by working on an independent project or seminar-type task.

Acquisition of Skills

When taking the exam during the exam period, there is a written part of the exam where tasks are solved, and an oral part of the exam, which, depending on the student's preference, can be a traditional oral exam or a presentation of the seminar paper.

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Homeworks 0 % 30 % 0 % 0 %
Class participation 0 % 10 % 0 % 0 %
Seminar/Project 0 % 30 % 0 % 30 %
Mid Term Exam: Written 0 % 35 % 0 %
Final Exam: Written 0 % 35 %
Final Exam: Oral 10 %
Exam: Written 0 % 70 %
Exam: Oral 30 %
Comment:

Up to 70% of the grade can be earned through a midterm and final exam (continuous teaching) or through a written exam (exam periods). The remaining up to 30% of the grade can be earned through the following activities (students can choose one or more of the offered activities, but the total points from these activities cannot exceed 30%): 1. Homework assignments (up to 30% of the grade) – this option is only available for students taking the course through continuous teaching, 2. Project (up to 30% of the grade), 3. Seminar (up to 30% of the grade). Homework assignments will involve solving tasks that deepen the knowledge acquired in the lectures, and they will also serve as preparation for the midterm and final exams. The project allows for the analysis and simulation of a mathematical model from engineering practice, while the seminar involves analyzing a selected scientific paper in the field of mathematical modeling. Particularly active students in class can earn an additional up to 10% of the grade through the Participation in Class component. Additionally, students who wish to improve the grade based on the points collected through the methods described for continuous teaching can do so through an optional oral exam, where they can earn up to 10% of the grade. During the exam periods, the oral exam (worth up to 30% of the grade) can be a traditional oral exam or a presentation of the seminar or project.

Week by Week Schedule

  1. Introduction to mathematical modeling. Dimensional analysis and applications.
  2. Calculation of the energy released in the Trinity test. Characteristic quantities and model analysis.
  3. Rocket launch in a varying gravitational field. Introduction to Finite Difference Method (FDM).
  4. Regular and singular perturbations.
  5. Modeling the spread of an epidemic (SIR model). State space, equilibrium points, and linearization of the model.
  6. Herd immunity. Advanced epidemic models.
  7. Predator-Prey models.
  8. Midterm
  9. Modeling of continua. Conservation law. Empirical laws of behavior. Diffusion and heat propagation.
  10. Separation of variables. Fundamental solution and superposition.
  11. Finite difference method for the heat diffusion equation. Reaction and diffusion equations.
  12. Industrial model: steel casting.
  13. Industrial model: water filtration.
  14. Industrial model: laser drill.
  15. Final Exam

Literature

Thomas Witelski, Mark Bowen (2015.), Methods of Mathematical Modelling: Continuous Systems and Differential Equations, Springer
David J. Wollkind, Bonni J. Dichone (2018.), Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences: Theoretical Predictions Compared with Data, Springer
Glenn Fulford, Philip Broadbridge (2002.), Industrial Mathematics: Case Studies in the Diffusion of Heat and Matter, Cambridge University Press
Avner Friedman, Walter Littman (1994.), Industrial Mathematics: A Course in Solving Real-World Problems, SIAM

General

ID 284077
  Winter semester
5 ECTS
L1 e-Learning
30 Lectures
0 Seminar
15 Exercises
0 Laboratory exercises
0 Project laboratory
0 Physical education excercises

Grading System

85 Excellent
75 Very Good
60 Good
50 Sufficient

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