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3 hrs agoMathematical Models – Explanation and Overview
A mathematical model is a way of describing a real-world situation using mathematical language, such as equations, functions, graphs, or algorithms. These models help scientists, engineers, and researchers analyze, predict, and understand how systems behave under different conditions.
🔹 Definition
A mathematical model is a representation of a system using mathematical concepts and relationships to describe how variables interact.
Formally:
Model: M=f(x1,x2,...,xn)\text{Model: } M = f(x_1, x_2, ..., x_n)Model: M=f(x1,x2,...,xn)
where x1,x2,...,xnx_1, x_2, ..., x_nx1,x2,...,xn are input variables and fff is a mathematical function that predicts or explains outcomes.
🔹 Types of Mathematical Models
Deterministic Models
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F=maF = maF=maNo randomness involved.
Same input → always same output.
Example: Newton’s laws of motion.
Stochastic (Probabilistic) Models
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Xt+1=aXt+ϵtX_{t+1} = aX_t + \epsilon_tXt+1=aXt+ϵtInclude elements of chance or uncertainty.
Example: Predicting stock prices or weather.
where ϵt\epsilon_tϵt is a random error term.
Static Models
Describe systems at one point in time.
Example: Supply-demand equilibrium model.
Dynamic Models
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dPdt=rP(1−PK)\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)dtdP=rP(1−KP)Show how systems change over time.
Example: Population growth.
Linear vs Nonlinear Models
Linear: Variables appear to the first power.
y=mx+by = mx + by=mx+bNonlinear: Variables have powers, products, or complex interactions.
y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c🔹 Examples of Mathematical Models
🔹 Steps in Building a Mathematical Model
Define the Problem – What system or process are we modeling?
Make Assumptions – Simplify the real world to focus on key factors.
Formulate Equations – Express relationships mathematically.
Solve the Model – Use analytical or numerical methods.
Validate the Model – Compare with real-world data.
Use the Model – Make predictions, optimize, or simulate.
🔹 Importance of Mathematical Models
Predict future outcomes (e.g., weather, disease spread).
Optimize systems (e.g., cost, energy, time).
Test scenarios safely (e.g., drug dosage, flight paths).
Improve decision-making using quantitative analysis.