Numerical Analysis

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The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306

Chapter 1

Mathematical Modeling
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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

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Mathematical Modeling

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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)

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Complex Mathematical Model
Example: Newton’s Second Law

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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

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Analytical solution to Newton's Second Law
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Analytical solution to Newton's Second Law

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Analytical solution to Newton's Second Law

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Numerical Solution to Newton's Second Law


Numerical solution: approximates the exact solution by arithmetic operations. Suppose 



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Numerical Solution to Newton's Second Law
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Numerical Solution to Newton's Second Law
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Comparison between...
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