We neither fear complexity nor embrace it for its own sake, but rather face it with the faith that simplicity and understanding are within reach. (Fred Adler)
Let us consider a straightforward scenario: From the standing crops, the Department of Agriculture seeks to determine the rice output in India. By cutting and weighing crops from a few typical fields, scientists may decide which regions are used for rice cultivation and the average production per acre. Decisions on the average rice yield are made using several statistical methodologies. How may mathematicians assist in resolving issues like these? They consult with specialists in the field and devise a mathematical solution to the problem. This equivalent consists of one or more equations, inequalities, and so on, known as mathematical models. They then solve the model and interpret the results in light of the original problem.
What is Mathematical Modelling?
Mathematical modelling is an effective tool that may be used in many fields, including biology, economics, and physics. Several examples of huge mathematical models with potentially significant effects on us are general weather forecasts, global warming, flight simulation, hurricane forecasting, nuclear winter, and nuclear arms race. In addition, mathematical models are used to simulate climate change scenarios, predict the spread of illnesses, and represent traffic flow, stock market choices, and predator-prey relationships. We may study the complexities of the world around us and influence the direction of technological growth through mathematical models.
Principles of Mathematical Modelling
A mathematical model is a simplified illustration of a real-world situation that incorporates assumptions and approximations. Obviously, the most crucial concern is whether our model is good, that is, whether the acquired results provide plausible responses when physically analysed. If a model needs to be sufficiently accurate, we attempt to discover the sources of the flaws. It is possible that we would need a new formulation, mathematical manipulation, and thus a new evaluation.
Mathematical modelling utilises ideas and methodologies to solve real-world problems. These principles are broad concerns regarding the objectives and purposes of modelling, similar to philosophical notions. They are documented in a series of questions and answers that theoretically guide the modelling process:
Why: What are we seeking for? Determine the necessity for the model.
Find: What do we want to know? List all the details that we're searching for.
Given: What do we know? Determine the available relevant data.
Assume: What assumptions could we make? Determine the situations that apply.
How: How should we view this model? Identify the governing physical principles.
Predict: What will the model predict? Determine the equations that will be utilised, the calculations that will be performed, and the results that will be obtained.
Valid: Are the predictions accurate? Identify tests that can be used to validate the model, such as whether it adheres to its principles and assumptions.
Verified: Are the predictions accurate? Identify tests that can be used to verify the model, i.e., is it helpful for the original reason it was created?
Improve: Can we enhance the model? Determine whether parameter values are inadequately understood, variables that should have been included, and/or assumptions/restrictions that can be lifted. Implement the iterative loop known as "model-validate-verify-improve-predict."
Use: How will we test the model? What shall we do with the model?
This set of questions and instructions does not constitute an algorithm for developing a useful mathematical model. However, its concepts are critical for modelling and problem-solving. As a result, we should expect to meet these specific questions frequently throughout the modelling process. This list should be considered as a generic approach to thinking about mathematical modelling.
Examples of Mathematical Modelling
Astrophysics relies heavily on mathematical models to comprehend celestial occurrences. For example, models of star formation, galaxy evolution, and black holes rely significantly on mathematical equations to explain observations and forecast future outcomes. This math modelling example focuses on some of the algebra behind the scenes: how to direct spacecraft, which aids scientists in unravelling the secrets of the universe.
Assume a corporation needs a computer for an extended period. The corporation can rent a computer for Rs. 2,000 per month or purchase one for Rs. 25,000. If the corporation has to use the computer for an extended period, it will pay such a high rent that buying a computer will be more cost-effective. However, hiring one will be less expensive if the organisation only needs the computer for a month. Determine the number of months after which it will be cheaper to purchase a computer. Let's think of a solution to this challenge. Suppose that it will be cheaper to buy a computer in x months. Hiring a computer for x months will cost Rs. 2,000 x. If the computer cost is less than the hiring costs (25/2 < x), the company will not be required to pay additional fees. The minimum value of x is 13 months. If the organisation has to hire a computer for 13 months or more, buying one will be less expensive.
Mathematical modelling is helpful in urban planning, particularly for regulating traffic flow. Models that replicate traffic patterns aid in the construction of efficient road networks, traffic light optimisation, and public transport system planning. This math modelling example predicts traffic congestion and suggests changes by considering variables such as vehicle density, speed, traffic restrictions, and human behaviour.
Common Mistakes in Mathematical Modelling
Remember that models, including complicated simulations, are only approximate representations of reality. They do not represent reality itself. This should always be kept in mind while formulating forecasts. For instance, it is preferable to say "that my model for climate change predicts a rise of three degrees in temperature" rather than "that there will be a three-degree rise." When utilising a model, you should constantly be mindful of its limitations, uncertainties, and approximations. The best models have quantified their uncertainty.
Fictitious Thinking: Fictitious Thinking: Rather than addressing the problem because it is too difficult, it is a typical mistake in mathematics—and, in fact, most subjects—to transform a problem into something we can solve. As a result, we may find ourselves applying extremely complex mathematics—or even the incorrect kind of mathematics—to answer what is, in reality, the wrong problem.
The ill effects of the formula: Do not assume this will always be the case, and don't simplify the issue excessively to find a precise formula.
Conclusion
We have outlined the definition of mathematical modelling and a guiding methodology for creating a well-motivated model. You could wonder how much we should work to improve our model. Generally, to make it better, we have to include additional variables in our mathematical equations to account for all conceivable components in our model. This may result in a complicated model that is challenging to operate. A model ought to be somewhat easy to use. Two factors are balanced in a good model: Accuracy, or how near to reality it is and usability.
Mathematical modelling is still being pursued despite the difficulties raised since knowledge and comprehension drive this never-ending process. We get a little closer to solving the mysteries of the world around us with every discovery and technological advancement. The foundation of human progress is mathematical modelling, which allows us to discover new things, decipher the workings of nature, and shape the destiny of our world.
Author:- Dr. Simran Arora - Assistant Professor
M.Sc. Mathematics - (CDOE)
