Applied Mathematics > Computational Mathematics > Mathematical Modelling
Description:
Mathematical Modelling is an essential subfield within Computational Mathematics, itself a major component of Applied Mathematics. This area focuses on the formulation, analysis, and interpretation of mathematical representations of real-world phenomena. These representations, known as models, are crucial in understanding, predicting, and optimizing various processes across multiple disciplines, such as physics, biology, engineering, economics, and social sciences.
Core Concepts:
Model Formulation:
Mathematical modelling begins with the translation of a problem from the real world into a mathematical framework. This involves identifying the key variables and parameters, establishing relationships among them, and making certain assumptions to simplify the problem while retaining its essential characteristics. The fundamental goal is to develop a set of equations or algorithms that describe the system’s behavior effectively.Types of Models:
- Deterministic Models: These models assume that outcomes are precisely determined through known relationships. They typically involve ordinary differential equations (ODEs), partial differential equations (PDEs), or algebraic equations.
- Stochastic Models: These models incorporate randomness and uncertainty. They are often represented using probabilistic approaches and stochastic differential equations (SDEs).
Analysis and Solution Methods:
Once the model is formulated, it must be analyzed and solved, either analytically or numerically.- Analytical Methods: These involve deriving exact solutions using mathematical techniques. For example, solving a linear system of differential equations may yield an explicit formula for the state variables.
- Numerical Methods: Many real-world problems result in complex models that cannot be solved analytically. Numerical methods, such as finite difference methods, finite element methods, and Monte Carlo simulations, approximate solutions by iterative computation.
Model Verification and Validation:
- Verification: Ensuring that the model is implemented correctly and the algorithms used to solve the model are accurately executing their intended functions.
- Validation: Comparing the model predictions with empirical data to check if the model accurately represents the real-world system. This step may involve statistical analysis and sensitivity testing to evaluate the robustness of the model.
Mathematical Formulations:
Consider a simple population growth model as an example. The population \( P(t) \) at time \( t \) can be modelled using the differential equation:
\[ \frac{dP}{dt} = rP (1 - \frac{P}{K}) \]
where:
- \( r \) is the intrinsic growth rate,
- \( K \) is the carrying capacity of the environment.
This nonlinear ordinary differential equation (ODE) represents the logistic growth of a population, capturing the initial exponential growth that slows down as the population reaches the carrying capacity.
Applications:
Mathematical modelling has vast applications, including:
- Epidemiology: Modelling the spread of infectious diseases to predict outbreaks and inform public health interventions. The SIR model, which divides a population into susceptible (S), infected (I), and recovered (R) compartments, is a classic example.
- Engineering: Designing and optimizing structures, machinery, and processes. Fluid dynamics models, for example, are used to simulate airflow over an aircraft wing.
- Economics: Analysing financial markets, risk assessment, and economic policy impacts through econometric models and game theory.
Conclusion:
Mathematical Modelling is a vital tool in Computational Mathematics, enabling the rigorous analysis and understanding of complex systems. By developing, analyzing, and validating mathematical models, practitioners can derive meaningful insights and provide solutions to real-world problems, bridging the gap between theory and practice.