Numerical Analysis
The Islamic University of Gaza Faculty of Engineering Civil Engineering Department
Numerical Analysis ECIV 3306
Chapter 1
Mathematical Modeling
10:51:09 PM
Part one : Approximation and Errors
Specific Study Objectives
• Recognize the difference between analytical and numerical solutions. • Recognize the distinction between truncation and round-off errors. • Understand the concepts of significant figures, accuracy, and precision. • Recognize the difference between true relative error t, approximate relative error a, and acceptable s error
10:51:09 PM
Chapter 1: Mathematical Modeling
Mathematical Model
• A formulation or equation that expresses the essential features of a physical system or process in mathematical terms. • Generally, it can be represented as a functional relationship of the form
10:51:09 PM
Mathematical Modeling
10:51:09 PM
Simple Mathematical Model
Example: Newton’s Second Law (The time rate of change of momentum of a body is equal to the resultant force acting on it)
a = acceleration (m/s2) ….the dependent variable m = mass of the object (kg) ….the parameter representing a property of the system. f = force acting on the body (N)
10:51:09 PM
Complex Mathematical Model
Example: Newton’s Second Law
10:51:09 PM
Where: c = drag coefficient (kg/s), v = falling velocity (m/s)
Complex Mathematical Model
At rest: (v = 0 at t = 0), Calculus can be used to solve the equation
10:51:09 PM
Analytical solution to Newton's Second Law
.
10:51:09 PM
Analytical solution to Newton's Second Law
10:51:09 PM
Analytical solution to Newton's Second Law
10:51:09 PM
Numerical Solution to Newton's Second Law
Numerical solution: approximates the exact solution by arithmetic operations. Suppose
10:51:09 PM
Numerical Solution to Newton's Second Law
.
10:51:09 PM
Numerical Solution to Newton's Second Law
.
10:51:09 PM
Comparison between
Numerical Analysis ECIV 3306
Chapter 1
Mathematical Modeling
10:51:09 PM
Part one : Approximation and Errors
Specific Study Objectives
• Recognize the difference between analytical and numerical solutions. • Recognize the distinction between truncation and round-off errors. • Understand the concepts of significant figures, accuracy, and precision. • Recognize the difference between true relative error t, approximate relative error a, and acceptable s error
10:51:09 PM
Chapter 1: Mathematical Modeling
Mathematical Model
• A formulation or equation that expresses the essential features of a physical system or process in mathematical terms. • Generally, it can be represented as a functional relationship of the form
10:51:09 PM
Mathematical Modeling
10:51:09 PM
Simple Mathematical Model
Example: Newton’s Second Law (The time rate of change of momentum of a body is equal to the resultant force acting on it)
a = acceleration (m/s2) ….the dependent variable m = mass of the object (kg) ….the parameter representing a property of the system. f = force acting on the body (N)
10:51:09 PM
Complex Mathematical Model
Example: Newton’s Second Law
10:51:09 PM
Where: c = drag coefficient (kg/s), v = falling velocity (m/s)
Complex Mathematical Model
At rest: (v = 0 at t = 0), Calculus can be used to solve the equation
10:51:09 PM
Analytical solution to Newton's Second Law
.
10:51:09 PM
Analytical solution to Newton's Second Law
10:51:09 PM
Analytical solution to Newton's Second Law
10:51:09 PM
Numerical Solution to Newton's Second Law
Numerical solution: approximates the exact solution by arithmetic operations. Suppose
10:51:09 PM
Numerical Solution to Newton's Second Law
.
10:51:09 PM
Numerical Solution to Newton's Second Law
.
10:51:09 PM
Comparison between