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Mathematical Modelling and Applications

Subject: Mathematics
Topic: 10
Cambridge Code: 0580

Introduction to Mathematical Modelling​

Mathematical model - Representation of real-world situation using mathematics

Modelling Process​

1. Identify the problem

  • What are we trying to understand/predict?
  • What are the constraints?
  • What factors are important?

2. Make assumptions

  • Simplify reality
  • Identify variables
  • State relationships
  • Specify domain

3. Translate to mathematics

  • Equations, functions, or systems
  • Appropriate mathematical tools

4. Solve mathematically

  • Algebraic, calculus, numerical methods
  • Rigorous mathematical process

5. Interpret results

  • Translate back to real world
  • Check reasonableness
  • Communicate findings

6. Validate model

  • Compare to real data
  • Refine if needed
  • State limitations

Types of Models​

Linear Models​

Form: y=mx+cy = mx + c

Applications:

  • Cost and revenue functions
  • Temperature vs time
  • Distance vs time (constant velocity)

Advantages:

  • Simple
  • Easy to interpret and solve

Limitations:

  • Real relationships often nonlinear
  • May only valid in limited range

Exponential Models​

Form: y=aekxy = ae^{kx} or y=abxy = ab^x

Applications:

  • Population growth
  • Radioactive decay
  • Bacterial growth
  • Compound interest

Exponential growth: k > 0 Exponential decay: k < 0

Polynomial Models​

Form: y=axn+bxn−1+...+cy = ax^n + bx^{n-1} + ... + c

Applications:

  • Projectile motion (quadratic)
  • Volume optimization
  • Cost functions

Power Models​

Form: y=axby = ax^b

Applications:

  • Relationship between body mass and height
  • Relationship between force and distance

Trigonometric Models​

Form: y=asin⁡(bx+c)+dy = a\sin(bx + c) + d or cosine variant

Applications:

  • Seasonal patterns
  • Oscillating systems
  • Circular motion
  • Sound waves

Optimization​

Single-Variable Optimization​

Objective: Maximize or minimize function

Method:

  1. Define function f(x)
  2. Find f'(x)
  3. Set f'(x) = 0
  4. Solve for critical points
  5. Use second derivative or context to determine max/min
  6. Check boundary values

Example: Minimize cost function C(x)=2x2−100x+1000C(x) = 2x^2 - 100x + 1000 C′(x)=4x−100=0C'(x) = 4x - 100 = 0 x=25x = 25 C′′(x)=4>0C''(x) = 4 > 0, so minimum at x = 25

Constrained Optimization​

Lagrange multipliers - Optimize subject to constraint

Maximize f(x,y) subject to g(x,y) = 0: ∇f=λ∇g∇f = λ∇g

Linear Programming​

Maximize/minimize: Linear objective function Subject to: Linear constraints

Method:

  1. Graph feasible region
  2. Identify vertices
  3. Evaluate objective at each vertex
  4. Optimal value at vertex (usually)

Growth and Decay​

Exponential Growth​

N(t)=N0ektN(t) = N_0 e^{kt} (continuous) N(t)=N0rtN(t) = N_0 r^t (discrete)

Half-life: Time for N to become N₀/2

Doubling time: Time for N to become 2N₀

Logistic Growth​

dNdt=rN(1−NK)\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Where K is carrying capacity

Solution: N(t)=K1+Ae−rtN(t) = \frac{K}{1 + Ae^{-rt}}

Applications:

  • Population under resource constraints
  • Disease spread with immunity
  • Adoption of new technology

Differential Equations in Modelling​

Separable Equations​

dydx=f(x)g(y)\frac{dy}{dx} = f(x)g(y)

Separate variables and integrate

First-Order Linear​

dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)

Solution using integrating factor

Applications​

Newton's Law of Cooling: dTdt=−k(T−Tambient)\frac{dT}{dt} = -k(T - T_{\text{ambient}})

Radioactive Decay: dNdt=−λN\frac{dN}{dt} = -λN

Mixing Problems: Rates of inflow and outflow

Approximation Methods​

Taylor Series​

f(x)≈f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+...f(x) ≈ f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ...

Linear approximation: f(x)≈f(a)+f′(a)(x−a)f(x) ≈ f(a) + f'(a)(x-a)

Numerical Methods​

Newton-Raphson: Find roots xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Euler's method: Solve differential equations yn+1=yn+hf(xn,yn)y_{n+1} = y_n + hf(x_n, y_n)

Trapezoid rule: Approximate integrals

Data Fitting and Regression​

Linear Regression​

Best fit line: Minimize sum of squared residuals y^=a+bx\hat{y} = a + bx

Polynomial Regression​

Fit polynomial: y^=a+bx+cx2+...\hat{y} = a + bx + cx^2 + ...

Choose degree: Balance fit vs simplicity

Exponential Regression​

Transform to linear: ln⁡(y)=ln⁡(a)+bx\ln(y) = \ln(a) + bx

Then apply linear regression

Model Validation​

Testing Against Data​

  1. Gather actual data
  2. Compare predictions to reality
  3. Calculate error measures:

Mean Absolute Error (MAE): MAE=1n∑∣yi−y^i∣\text{MAE} = \frac{1}{n}\sum |y_i - \hat{y}_i|

Root Mean Square Error (RMSE): RMSE=1n∑(yi−y^i)2\text{RMSE} = \sqrt{\frac{1}{n}\sum (y_i - \hat{y}_i)^2}

Parameter Estimation​

From data:

  • Fit model to existing observations
  • Use statistical methods
  • Cross-validation for reliability

Common Modelling Scenarios​

Population Models​

Discrete: Pn+1=Pn+rPn=Pn(1+r)P_{n+1} = P_n + rP_n = P_n(1 + r)

Continuous: dPdt=rP\frac{dP}{dt} = rP

With carrying capacity: Logistic model

Economic Models​

Revenue: R = price × quantity Cost: Fixed + variable costs Profit: Revenue - Cost Break-even: Profit = 0

Motion Problems​

Position, velocity, acceleration: v=dsdt,a=dvdtv = \frac{ds}{dt}, \quad a = \frac{dv}{dt}

Projectile motion: h(t)=h0+v0sin⁡(θ)t−12gt2h(t) = h_0 + v_0\sin(θ)t - \frac{1}{2}gt^2

Optimization in Business​

Production quantity optimizing profit Inventory levels balancing cost and availability Pricing maximizing revenue

Limitations and Assumptions​

Common Assumptions​

  • Variables continuous vs discrete
  • Linear vs nonlinear relationships
  • Constant vs changing parameters
  • Independence of variables
  • Normal distribution of errors

Model Limitations​

  • Based on assumptions
  • Limited to specific conditions
  • Simplifies reality
  • Prediction uncertainty
  • Extrapolation dangers

Communicating Uncertainty​

  • Confidence intervals
  • Sensitivity analysis
  • Scenario analysis
  • Stated assumptions

Key Points​

  1. Modelling involves idealization of reality
  2. Choose appropriate mathematical form
  3. Exponential for growth/decay phenomena
  4. Logistic for growth with constraints
  5. Optimization uses calculus
  6. Differential equations model change
  7. Validate models with real data
  8. Calculate errors to assess fit
  9. State assumptions and limitations
  10. Communicate results clearly

Practice Questions​

  1. Set up models for real scenarios
  2. Estimate parameters from data
  3. Solve optimization problems
  4. Solve differential equations
  5. Apply numerical methods
  6. Fit models to data
  7. Calculate error measures
  8. Validate models
  9. Compare alternative models
  10. Predict future values

Revision Tips​

  • Understand modelling process thoroughly
  • Practice setting up equations
  • Know when to use each model type
  • Understand assumptions matter
  • Practice optimization techniques
  • Work with real data
  • Validate conclusions
  • Communicate assumptions
  • Understand limitations
  • Develop problem-solving skills