Mathematical Modelling

Hey students! 👋 Welcome to one of the most exciting and practical areas of A-level Further Mathematics - Mathematical Modelling! This lesson will teach you how to transform real-world problems into mathematical equations that we can actually solve. You'll learn the essential skills of model formulation, making smart assumptions, nondimensionalisation, linearisation, and validating your models against real data. By the end of this lesson, you'll be able to tackle complex problems like population growth, pendulum motion, and economic systems using the power of mathematics! 🚀

Understanding Mathematical Modelling and Assumptions

Mathematical modelling is the art and science of translating real-world situations into mathematical language. Think of it as building a bridge between the messy, complicated real world and the clean, precise world of mathematics. When you model a situation, you're essentially creating a simplified version of reality that captures the most important features while ignoring less significant details.

The first and most crucial step in mathematical modelling is making appropriate assumptions. These assumptions are like filters that help you decide what to include and what to leave out of your model. For example, when modelling the motion of a falling ball, you might assume that air resistance is negligible for small speeds, or that the acceleration due to gravity is constant near Earth's surface (approximately 9.81 m/s²).

Let's consider a classic example: population growth. If we want to model how a population of bacteria grows in a petri dish, we might start with some basic assumptions. We could assume that the population grows at a rate proportional to the current population size, leading to the simple exponential model: $\frac{dP}{dt} = rP$ where P is the population size, t is time, and r is the growth rate constant. However, this model predicts unlimited growth, which isn't realistic! 📈

A more sophisticated approach considers that resources become limited as the population grows. This leads us to the famous logistic growth model: $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ where K represents the carrying capacity of the environment. This model shows how populations initially grow exponentially but then level off as they approach their environmental limits.

The key to good mathematical modelling lies in making assumptions that are realistic enough to capture the essential behavior of the system, but simple enough that the resulting mathematics remains tractable. This requires both mathematical skill and physical intuition about the problem you're trying to solve.

Nondimensionalisation: Making Models Universal

Nondimensionalisation is a powerful technique that transforms your equations by removing units and reducing the number of parameters. This might sound abstract, but it's incredibly useful! When you nondimensionalise a model, you're essentially finding the "natural" scales of the problem and expressing everything in terms of these scales.

Let's work through this with the logistic growth model. Our dimensional equation is: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$

To nondimensionalise, we introduce dimensionless variables. Let's define $u = \frac{P}{K}$ (so u represents the population as a fraction of the carrying capacity) and $\tau = rt$ (a dimensionless time). Using the chain rule, we get: $$\frac{du}{d\tau} = u(1-u)$$

Amazing! 🎯 We've transformed our equation from having three parameters (P, r, and K) to having no parameters at all! This nondimensional form tells us that all logistic growth processes, whether they're bacteria in a petri dish or rabbits in Australia, follow the same mathematical pattern when expressed in their natural scales.

Nondimensionalisation offers several benefits. First, it reduces the number of parameters you need to consider, making the mathematics simpler. Second, it reveals the fundamental structure of the problem, showing you which combinations of parameters are truly important. Third, it makes your results more general - the nondimensional solution applies to any logistic growth process, regardless of the specific values of r and K.

Consider another example: the motion of a pendulum. The dimensional equation for small oscillations is: $\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0$ where θ is the angle, g is gravitational acceleration, and L is the pendulum length. By introducing dimensionless time $\tau = t\sqrt{\frac{g}{L}}$, we get: $$\frac{d^2\theta}{d\tau^2} + \sin\theta = 0$$

This shows us that all pendulums, regardless of their length, behave identically when time is measured in units of $\sqrt{\frac{L}{g}}$!

Linearisation: Simplifying Complex Behavior

Real-world systems often involve nonlinear relationships that can be mathematically challenging to solve exactly. Linearisation is a technique that approximates nonlinear systems with linear ones, making them much easier to analyze. This approach is particularly powerful when you're interested in understanding behavior near equilibrium points or for small perturbations from a known state.

The mathematical foundation of linearisation is the Taylor series expansion. For a function f(x) near a point x₀, we can write: $$f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + ...$$

For linearisation, we keep only the first-order terms, giving us: $$f(x) \approx f(x_0) + f'(x_0)(x-x_0)$$

Let's apply this to our pendulum example. The exact equation is: $$\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0$$

For small angles, we can linearise by approximating sin θ ≈ θ (since sin θ = θ - θ³/6 + ... and for small θ, the higher-order terms become negligible). This gives us: $$\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$$

This linear equation has the well-known solution θ(t) = A cos(ωt + φ), where $\omega = \sqrt{\frac{g}{L}}$ is the natural frequency, and A and φ are determined by initial conditions. The linearised model predicts that small pendulum oscillations are perfectly harmonic with a period of $T = 2\pi\sqrt{\frac{L}{g}}$ - a result that matches experimental observations very well for small angles! ⏰

Linearisation is also crucial in understanding stability. Consider the logistic growth model again: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$

This system has two equilibrium points: P = 0 and P = K. To understand their stability, we linearise around each point. Near P = 0, if we let P = 0 + ε (where ε is small), we get: $$\frac{d\varepsilon}{dt} \approx r\varepsilon$$

This shows that small perturbations from P = 0 grow exponentially, making this equilibrium unstable. Near P = K, letting P = K + ε gives: $$\frac{d\varepsilon}{dt} \approx -r\varepsilon$$

Here, small perturbations decay exponentially, making P = K a stable equilibrium. This analysis explains why populations tend to approach their carrying capacity over time.

Model Validation: Testing Against Reality

Creating a mathematical model is only half the battle - you must also validate it against real data to ensure it actually describes the system you're studying. Model validation is the process of comparing your model's predictions with experimental observations or real-world data to assess its accuracy and reliability.

There are several approaches to validation. The most direct method is to use your model to make predictions and then compare these predictions with new experimental data. For example, if you've developed a model for population growth, you could use it to predict population sizes at future times and then check whether these predictions match actual measurements.

Statistical measures help quantify how well your model fits the data. The coefficient of determination (R²) measures the proportion of variance in the data that your model explains, with values closer to 1 indicating better fits. The root mean square error (RMSE) quantifies the average difference between predicted and observed values: $RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}$ where yᵢ are the observed values and ŷᵢ are the predicted values.

Let's consider a real example: validating the logistic growth model for yeast populations. Researchers have found that when yeast is grown in a controlled environment with limited nutrients, the population follows the logistic curve remarkably well. Data from laboratory experiments show that yeast populations initially grow exponentially, then gradually slow down as they approach the carrying capacity of their environment. The R² values for logistic fits to yeast data often exceed 0.95, indicating excellent agreement between model and reality! 🧪

However, validation also involves understanding when and why models fail. The simple pendulum model works excellently for small angles but becomes increasingly inaccurate as angles increase. For a pendulum with an initial angle of 90°, the linearised model predicts a period that's about 18% shorter than the actual period. This doesn't mean the model is "wrong" - it means we understand its limitations and can use it appropriately.

Cross-validation is another important technique. This involves splitting your data into training and testing sets. You use the training data to determine model parameters, then test how well the model predicts the testing data. This approach helps identify overfitting, where a model performs well on the data used to create it but poorly on new data.

Conclusion

Mathematical modelling is a powerful toolkit that allows you to tackle complex real-world problems using mathematical techniques. The process involves making thoughtful assumptions to simplify reality, using nondimensionalisation to reveal fundamental structures, applying linearisation to make problems tractable, and validating results against experimental data. These skills work together to transform abstract mathematical concepts into practical tools for understanding everything from population dynamics to engineering systems. Remember, the goal isn't to create perfect models, but rather useful ones that provide insight and make accurate predictions within their range of validity.

Study Notes

• Mathematical modelling transforms real-world problems into mathematical equations through careful assumptions and simplifications

• Key assumptions should capture essential behavior while keeping mathematics tractable (e.g., neglecting air resistance for slow-moving objects)

• Logistic growth model: $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$ where P is population, r is growth rate, K is carrying capacity

• Nondimensionalisation removes units and reduces parameters by introducing dimensionless variables scaled to natural problem scales

• Linearisation approximates nonlinear functions using Taylor series: $f(x) \approx f(x_0) + f'(x_0)(x-x_0)$ for small deviations from $x_0$

• Small angle approximation: $\sin\theta \approx \theta$ for linearising pendulum motion: $\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$

• Pendulum period: $T = 2\pi\sqrt{\frac{L}{g}}$ for small oscillations

• Stability analysis uses linearisation around equilibrium points to determine if small perturbations grow or decay

• Model validation compares predictions with experimental data using statistical measures like R² and RMSE

• RMSE formula: $RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}$ where $y_i$ are observed and $\hat{y}_i$ are predicted values

• Cross-validation splits data into training and testing sets to check for overfitting and assess predictive power