Dynamic Mathematical Model

Definition

A mathematical model uses symbolic notation and mathematical equations to represent a system. The system attributes are represented by variables and the system activities are represented by mathematical functions.

A dynamic mathematical model is a mathematical model in which the system attributes (variables) change with time, i.e., the model is a function of time. The state of the system at any given moment depends on previous states and the passage of time. Dynamic models capture the behavior and evolution of a system over time.

In simple terms: a dynamic mathematical model describes how a system changes from one state to another as time progresses.

Example of Dynamic Mathematical Model

Example 1: Population Growth Model

The population of a city changes with time. This can be modeled as:

$$\frac{dP}{dt} = rP$$

Where:

• $P$ = population at time $t$
• $r$ = growth rate constant
• $t$ = time

The solution gives:

$$P(t) = P_0 \cdot e^{rt}$$

Here, $P$ is a function of time $t$, so the value of the attribute (population) changes continuously. This is a dynamic mathematical model.

Example 2: Simple Harmonic Motion (Spring-Mass System)

$$m\frac{d^2x}{dt^2} + kx = 0$$

Where:

• $m$ = mass
• $k$ = spring constant
• $x$ = displacement (changes with time)

The displacement $x$ changes with time, making this a dynamic mathematical model.

Example 3: Bank Account with Interest

$$A(t) = P\left(1 + \frac{r}{n}\right)^{nt}$$

The amount $A$ changes as time $t$ increases, representing a dynamic behavior.

Differentiation: Dynamic vs Static Mathematical Model

Summary

• A static mathematical model gives a fixed relationship between variables (e.g., $y = mx + c$), where the system does not evolve with time.
• A dynamic mathematical model captures how a system evolves over time using differential equations or time-dependent functions (e.g., $\frac{dP}{dt} = rP$).
• Dynamic models are essential in simulation because most real-world systems (weather, population, economics, engineering systems) are inherently time-varying.

Monte Carlo Simulation Method

Definition

Monte Carlo simulation is a computerized mathematical technique that generates random sample data based on some known distribution for numerical experiments. It is applied to risk quantitative analysis and decision making problems, and is used by professionals in finance, project management, energy, manufacturing, engineering, research & development, transportation, and more.

Key Characteristics

The three important characteristics of the Monte Carlo method are:

1. The output must generate random samples
2. The input distribution must be known
3. The result must be known while performing an experiment

Flowchart for Monte Carlo Simulation

        Start
          |
          v
  Define the Problem
          |
          v
  Identify Input Variables
  and their Distributions
          |
          v
  Generate Random Numbers
          |
          v
  Simulate the Experiment
  (Repeat N times)
          |
          v
  Collect and Analyze Output
          |
          v
  Estimate the Result / Decision
          |
          v
        Stop

Example: Estimating the Value of Pi (π)

Problem: Use Monte Carlo simulation to estimate the value of π.

Concept:

• Consider a unit square of side 1 and a quarter circle of radius 1 inscribed within it.
• Area of the square = 1 × 1 = 1
• Area of the quarter circle = π(1)²/4 = π/4

Steps:

Step 1: Generate N pairs of random numbers (x, y) where x, y ∈ [0, 1]

Step 2: Check if the point (x, y) falls inside the quarter circle using:

$$x^2 + y^2 \leq 1$$

Step 3: Count the number of points inside the circle (M)

Step 4: Estimate π using:

$$\frac{\text{Points inside circle}}{\text{Total points}} = \frac{\pi/4}{1}$$

$$\pi \approx 4 \times \frac{M}{N}$$

Numerical Illustration:

• Total points N = 5
• Points inside circle M = 3

$$\pi \approx 4 \times \frac{3}{5} = 4 \times 0.6 = \mathbf{2.4}$$

Note: As N increases (e.g., N = 10,000), the estimate converges closer to the true value of π ≈ 3.14159.

Advantages

• Can handle complex systems that are difficult to solve analytically
• Applicable to a wide range of domains
• Provides probabilistic results with confidence intervals

Summary

Monte Carlo simulation uses repeated random sampling to obtain numerical results. The larger the number of trials, the more accurate and reliable the result becomes. It is especially powerful when analytical solutions are difficult or impossible to obtain.

Multiplicative Congruential Method

Step 1 - Given Data

Step 2 - Solve

Recurrence: $X_{n+1} = (a \cdot X_n) \bmod m = (11 \cdot X_n) \bmod 16$

Iteration 1: $$X_1 = (11 \times 7) \bmod 16 = 77 \bmod 16 = 13 \quad (77 = 4\times16 + 13)$$ $$R_1 = \frac{13}{16} = 0.8125$$

Iteration 2: $$X_2 = (11 \times 13) \bmod 16 = 143 \bmod 16 = 15 \quad (143 = 8\times16 + 15)$$ $$R_2 = \frac{15}{16} = 0.9375$$

Iteration 3: $$X_3 = (11 \times 15) \bmod 16 = 165 \bmod 16 = 5 \quad (165 = 10\times16 + 5)$$ $$R_3 = \frac{5}{16} = 0.3125$$

Iteration 4: $$X_4 = (11 \times 5) \bmod 16 = 55 \bmod 16 = 7 \quad (55 = 3\times16 + 7)$$ $$R_4 = \frac{7}{16} = 0.4375$$

Iteration 5: $$X_5 = (11 \times 7) \bmod 16 = 77 \bmod 16 = 13$$

Since $X_5 = X_1 = 13$, the sequence repeats. Period = 4.

Summary Table

Result

Sequence of integers: $13, 15, 5, 7, 13, \ldots$

Sequence of random numbers: $$R = {0.8125,\ 0.9375,\ 0.3125,\ 0.4375, \ldots}$$

The period is 4, short because $m = 16$ is small.