Introduction

Objectives

• Formulate mathematical models based on scientific principles for physical systems.

• Utilize numerical methods for solution generalization applicable on digital computers.

• Understand conservation laws in engineering models; recognize steady-state vs dynamic solutions.

• Explore various numerical methods.

Mathematical Model Overview

• Mathematical models express the essential features of a physical system.

• Represented by relationships involving dependent variables, independent variables, parameters, and forcing functions.

Components of Model Functions

• Dependent variable: Reflects the behavior/state of the system.

• Independent variables: Dimensions like time and space for system behavior analysis.

• Parameters: Constants defining system properties.

• Forcing functions: External influences affecting the system.

Standard Form of Model Functions

• Generic form for a first-order differential equation:
  $\frac{dy}{dt} = f(t, y, p, F)$

• $y$: Dependent variable (e.g., temperature, velocity)

• $t$: Independent variable (e.g., time, position)

• $p$: Parameters (e.g., system constants)

• $F$: Forcing functions (e.g., external influences)

Analytical vs Numerical Solutions

• Analytical solutions: Involve exact formula derivations for the state of the system, often expressed as $y(t)$.

• Numerical solutions: Use iterative calculations, such as discrete time-stepping, to approximate system behavior.

Euler's Method

• A numerical technique for approximating solutions of differential equations by iterating calculations in finite intervals.

• Standard Formula:
  $v<em>{i+1} = v</em>i + (\text{slope} \times \text{step})$

• Iterative Process: Calculate the next value by adding the product of the estimated slope and the step size to the current value.

Case Studies in Standard Form

• Newton's Law of Cooling:
  $\frac{dT}{dt} = -k(T - T_a)$

• Calculates the rate of temperature change relative to ambient temperature $T_a$.

• Evaporation of Liquid Droplet:
  $\frac{dV}{dt} = -k A$

• Models volume reduction as proportional to the surface area $A$ of the droplet.

• Fluid in Storage Tank:
  $\frac{dh}{dt} = \frac{Q<em>{in} - Q</em>{out}}{A}$

• Solves for changes in liquid depth $h$ based on inflow $Q<em>{in}$ and outflow $Q</em>{out}$ rates relative to cross-sectional area $A$.