Mathematical Modelling In Real Life Problems - Unveiling the Secrets

Mathematical modelling is a powerful tool that allows us to understand, analyze, and solve complex real-life problems through the application of mathematical concepts and techniques. From predicting the weather to optimizing traffic flow, mathematical modelling plays a crucial role in various fields, including science, engineering, finance, and even social sciences. In this article, we will delve into the fascinating world of mathematical modelling and explore how it helps us tackle real-life problems.

The Basics of Mathematical Modelling

To begin with, let's first understand what mathematical modelling actually means. Mathematical modelling involves the process of formulating a mathematical representation of a real-life problem. This representation, known as a mathematical model, consists of equations, variables, and parameters that describe the problem's essential characteristics and relationships.

Mathematical modelling typically follows these key steps:

Mathematical Modelling in Real Life Problems: Case Studies from ECMI-Modelling Weeks (Mathematics in Industry Book 33)

by James A. B. Mahaffey Jr. (1st ed. 2020 Edition, Kindle Edition)

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Language	:	English
File size	:	5858 KB
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Print length	:	173 pages

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1. Problem Identification: In this step, the real-life problem that needs to be solved using mathematical modelling is identified. It could be anything from optimizing a manufacturing process to designing an efficient transportation system.
2. Model Formulation: Once the problem is identified, we need to create a mathematical model that represents the problem accurately. This involves deciding what variables and parameters to include and formulating equations that describe their relationships.
3. Solution Techniques: After formulating the model, we employ various mathematical techniques, such as calculus, linear algebra, and probability theory, to solve the equations and obtain solutions. These solutions provide insights and answers to the real-life problem at hand.
4. Validation and Analysis: Once we have obtained the solutions, we validate the mathematical model by comparing its predictions with real-life data or existing experimental results. This step helps us assess the accuracy and reliability of the model.
5. Application: Finally, the mathematical model, along with its validated solutions, can be used to make predictions, optimize systems, or guide decision-making processes in real-life situations.

Applications of Mathematical Modelling

Mathematical modelling finds extensive applications across various disciplines. Here are a few notable examples:

1. Weather Prediction

To forecast the weather, meteorologists use mathematical models that simulate the atmosphere's behavior based on various parameters such as temperature, wind speed, and pressure. These models help predict weather patterns, track storms, and issue early warnings.

2. Epidemic Spread

During outbreaks of infectious diseases, mathematical models can assist in predicting the spread of the disease, estimating its impact on public health, and evaluating the effectiveness of control measures like vaccination campaigns and social distancing.

3. Financial Markets

Mathematical models are extensively used in finance to predict stock prices, analyze investment portfolios, and calculate risks. These models allow investors and financial institutions to make informed decisions and manage their assets effectively.

4. Traffic Flow Optimization

Mathematical models help simulate and optimize traffic flow, thereby improving transportation systems' efficiency. By considering factors like road capacity, traffic patterns, and signal timings, these models aid in identifying congestion-prone areas and developing effective traffic management strategies.

Challenges in Mathematical Modelling

While mathematical modelling is a powerful tool, it comes with its fair share of challenges. Some of these include:

• Data Availability: Obtaining accurate and reliable data for constructing and validating mathematical models can be a significant challenge, particularly for complex real-life problems.
• Assumptions: Modelling often requires making simplifying assumptions to make the problem tractable. However, these assumptions may not perfectly represent the real system under study, leading to potential inaccuracies.
• Complexity: Real-life problems are inherently complex, and capturing all relevant variables and relationships accurately in a mathematical model can be a daunting task.
• Validity of Models: It is essential to validate mathematical models thoroughly to ensure their reliability and applicability to real-life situations. However, validation can be challenging due to the dynamic nature of many real-life systems.

Mathematical modelling is an indispensable tool for understanding, analyzing, and solving real-life problems in various fields. By formulating mathematical models and employing appropriate techniques, we can gain valuable insights, make predictions, and optimize systems. However, it is important to recognize the challenges associated with mathematical modelling and exercise caution in interpreting and applying the results. With its immense potential, mathematical modelling continues to push the boundaries of human knowledge and improve our understanding of the world.

Mathematical Modelling in Real Life Problems: Case Studies from ECMI-Modelling Weeks (Mathematics in Industry Book 33)

by James A. B. Mahaffey Jr. (1st ed. 2020 Edition, Kindle Edition)

5 out of 5

Language	:	English
File size	:	5858 KB
Screen Reader	:	Supported
Print length	:	173 pages

FREE

This book is intended to be a useful contribution for the modern teaching of applied mathematics, educating Industrial Mathematicians that will meet the growing demand for such experts.

It covers many applications where mathematics play a fundamental role, from biology, telecommunications, medicine, physics, finance and industry. It is presented in such a way that can be useful in Modelation, Simulation and Optimization courses, targeting master and PhD students. Its content is based on many editions from the successful series of Modelling Weeks organized by the European Consortium of Mathematics in Industry (ECMI). Each chapter addresses a particular problem, and is written in a didactic way, providing the description of the problem, the particular way of approaching it and the proposed solution, along with the results obtained.