Engineering Analysis (650201) lec(1)-introduction

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1 Philadelphia University Faculty of Engineering Communication and Electronics Engineering Engineering Analysis (650201) lec(1)-introduction Instructor: Eng. Nada

2 Course Outline Course Title: Engineering Analysis (1) (650201) Prerequisite: Engineering Mathematics (210206) Text Book : Advanced Engineering Mathematics By: Erwin Kreyszig 8th edition Credit Hours :3 Level :3rd year Providing Dept. :Communications & Electronics Engineering Instructor :Nada N Khatib Office No: 730 lec 1-introduction Eng. Nada khatib 2

3 Course Contents Basic Concepts & Ideas (ch.1) First Order Differential Equations (ch.1) Second & Higher Order Differential Equations (ch. 2) Laplace transform (ch. 5) Power Series Method (ch. 4) Introduction to Partial Differential Equations lec 1-introduction Eng. Nada khatib 3

4 Text Book and References Advanced Engineering Mathematics By: Erwin Kreyszig 8th edition.1999 Boyce, William E., DiPrima, Richard C., Elementary Differential Equations, Fifth Edition, Wiley, New York, Rabenstein, Albert L., Elementary Differential Equations with Linear Algebra, Third Edition, Academic Press, New York, Krusemeyer, Mark, Differential Equations, Macmillan Publishing Co., New York, Elementary Differential Equations and Boundary Value Problems by W.E. Boyce R.C. Diprima. New York, NY, U.S.A. : John Wiley & Sons, Incorporated, الدكتور عبد الرحمن القواسمي و المهندسة ندى نبيل الخطيب " المعادالت التفاضلية: تطبيقاتها الهندسية" الطبعة االولى طبع بدعم من جامعة فيالدلفيا مطبعة الخط العربي lec 1-introduction Eng. Nada khatib 4

5 Course Grading First Exam (20%) Second Exam (20%) Quizzes and Homework (20%) Three homework assignments and 3 quizzes There is no late homework policy The grade of each Homework assignment is 8 The grades assigned to Quizzes will be elements of {0, 1, 2, 3, 4} Missed quizzes cannot be made-up Final Exam (40%) lec 1-introduction Eng. Nada khatib 5

6 What is the derivative? Definition 1 Definition 2 Definition 3 The rate of change of a function The slope of the tangent line to the graph of a function The best linear approximation of a function All of these answers are correct lec 1-introduction Eng. Nada khatib 6

7 What is the Differential Equation? Differential Equation (DE):is an equation containing the derivatives or differentials of one or more dependent variables, with respect to one or more independent variables. DEs are used to describe phenomena that are in movement or change with time lec 1-introduction Eng. Nada khatib 7

8 What is the Differential Equation? The DE may contains also the unknown function itself, as well as the given function and constants t 2 x x 3 t cos t 2 d x dt dx dt 9x 2cos 3t lec 1-introduction Eng. Nada khatib 8

9 Mathematical Model DE are of fundamental importance in engineering mathematics because many physical laws & relations appear mathematically in the form of such equations lec 1-introduction Eng. Nada khatib 9

10 Mathematical Model DEs describe (almost) everything in the world: physics chemistry engineering ecology economy weather lec 1-introduction Eng. Nada khatib 10

11 Modeling Modeling is the process of describing a physical situation with a DE It is the derivation of DEs from physical or other problems lec 1-introduction Eng. Nada khatib 11

12 Population Models How does one predict the growth of a population? Human populations Population of bacteria lec 1-introduction Eng. Nada khatib 12

13 Population Models/Human populations Growth rate is proportional to the population present Assume that people die only of natural causes Death rate is also proportional to the population present lec 1-introduction Eng. Nada khatib 13

14 Population Models/Human population if P(t) represents a population in a given region at any time t then Rate of = Rate at which - Rate at which change of P(t) enters the P(t) exits the P(t) region region dp p ( k k ) 1 2 dt p kp lec 1-introduction Eng. Nada khatib 14

15 Population dp dp dt kp p kdt ln p kt c e ln p e kt c kt p ce lec 1-introduction Eng. Nada khatib 15

16 Population Models/Population of bacteria Reproduce by simple cell division Parent cell does not die, but become two new cells Growth rate is proportional to the population present dp p k1 dt p lec 1-introduction Eng. Nada khatib 16

17 Heating and cooling of Building How long does it take to change the building temperature substantially? How does the building temperature vary during spring and fall when there is no furnace heating or air condition? lec 1-introduction Eng. Nada khatib 17

18 Heating and cooling of Building How does the building temperature vary in summer when there is air condition? How does the building temperature vary in winter when there is furnace heating? lec 1-introduction Eng. Nada khatib 18

19 Newton s law of cooling Newton s law of cooling, states that the temperature of a hot or a cold object decreases or increases at a rate proportional to the difference of the temperature of the object and that of its surrounding. lec 1-introduction Eng. Nada khatib 19

20 Temperature Problems dt dt k ( T Ts ) T= the temperature of a body at any time (inside temperature) T s = the temperature of the surrounding medium (outside temperature) lec 1-introduction Eng. Nada khatib 20

21 Temperature Problems dt dt k( T T ) H( t) U( t) s H= additional heating rate U= furnace heating (+ve) U= air condition cooling (-ve) lec 1-introduction Eng. Nada khatib 21

22 Pendulums Determine the displacement angle, measured from the vertical, as a function of time in pendulums lec 1-introduction Eng. Nada khatib 22

23 Pendulums The swinging bob in a grandfather s clock & A child s swing Are examples of pendulums lec 1-introduction Eng. Nada khatib 23

24 Mixing problems The mixing of a two fluids gives rise to a DE lec 1-introduction Eng. Nada khatib 24

25 Newton s 2 nd law of motion an object of mass (m) is moving with acceleration (a) that directly proportional to the total force (F) acting on it and inversely proportional to its mass m a F we can rewrite the acceleration, a, in one of two ways v ( t) a h ( t) a lec 1-introduction Eng. Nada khatib 25

26 Newton s 2 nd law of motion Newton s Second Law can now be written as a differential equation in terms of either the velocity, v, or the position mv ( t) mg mh ( t) mg lec 1-introduction Eng. Nada khatib 26

27 Newton s 2 nd law of motion To find the velocity of the body at any time in case of free-falling objects, we need to solve the DE mv ( t) mg lec 1-introduction Eng. Nada khatib 27

28 Newton s 2 nd law of motion To find the position of the body relative to the ground in case of freefalling objects, we need to solve the DE mh ( t) mg lec 1-introduction Eng. Nada khatib 28

29 Newton s 2 nd law of motion To find the velocity of the sky driver at any time lec 1-introduction Eng. Nada khatib 29

30 Mass-spring system The motion of a basic mechanical systems (a mass on an elastic spring) are governed by differential equations. lec 1-introduction Eng. Nada khatib 30

31 Satellite To determine the distance r, you should use Newton s 2 nd law and universal law of gravitation then solve DE. lec 1-introduction Eng. Nada khatib 31

32 Suspended Wire To determine the shape of a wire hanging under its own weight, such as a wire between two telephone line poles, we must solve nonlinear DE lec 1-introduction Eng. Nada khatib 32

33 Electric circuit The electric circuits (LR, RC, LRC) models are differential equations lec 1-introduction Eng. Nada khatib 33

34 Elastic Beam An important fourth order Differential equation governs the bending of an elastic beam such as wooden or iron girder in a building or a bridge lec 1-introduction Eng. Nada khatib 34

35 What is the Differential Equation? We can write any DE in either following forms: explicit form y ( n) ( n 1) f ( x, y, y, y,..., y ) lec 1-introduction Eng. Nada khatib 35

36 What is the Differential Equation? Implicit form F( y ( n), y ( n 1),..., y, y, y, x) 0 lec 1-introduction Eng. Nada khatib 36

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations MATH 210 Introduction to Differential Equations Fall 2016-2017 Instructor Room No. Office Hours Email Telephone Secretary/TA TA Office Hours Course URL (if any) Ali Ashher Zaidi ali.zaidi@lums.edu.pk Math.lums.edu.pk/moodle