Introduction Eigenfunction Expansions January 21 2009 Overview Course Introduction and Eigenfunction Expansions Course outline schedule grading exams homework office hours etc Problems considered Eigenfunction expansions Larry Caretto Mechanical Engineering 501B Why do we care about this Sturm Liouville problem Complete orthogonal eigenfunctions Examples of eigenfunction expansions Seminar in Engineering Analysis January 21 2009 Sines and cosines Bessel functions Other eigenfunctions Course Structure Course Materials ME 501AB is a one year course in engineering and numerical analysis Web site http www csun edu lcaretto me501b Lecture presentations supplemented in some cases by course notes Reading assignments in text should be done prior to class Download notes prior to class Homework assignments and solutions What s new section on home page for recent additions Review 501A topics when required Look at solving problems once they are formulated Two overall goals Understand advanced mathematical and computational approaches encountered in your work and future course work Develop the ability to apply appropriate problem solving skills 3 Instructor and Course Data lcaretto csun edu 818 677 6448 Extensive email availability Office JD 3333 office hours MW 5 6 pm and TTh 2 3 pm other times by drop in phone call or appointment Texts Kreyszig Advanced Engineering Mathematics and Hoffman Numerical Methods for Engineers and Scientists Grading based on homework 10 2 midterms 50 and a final 40 See grading criteria in outline 5 ME 501B Engineering Analysis 2 4 Course Objectives Understand and be able to apply analytical and numerical solutions of partial differential equations Read publications of applied engineering analysis and numerical analysis that involve ordinary and partial differential equations Related topics include matrix operations and special functions such as Bessel functions 6 1 Introduction Eigenfunction Expansions January 21 2009 Course objectives II Course objectives III Be familiar with algorithms and software packages for differential equations and understand their limitations Solve engineering problems using differential equations and understand when numerical solutions are required Understand differences between parabolic hyperbolic and elliptic equations in both analytical and numerical approaches Understand methods used in numerical analysis and be able to apply them in simple cases Finite difference approaches are simplest Finite element approaches are used for complex geometries Both convert differential equations to a system of algebraic equations that is solved using numerical approaches 7 Galileo Galilei 1564 1642 You cannot teach people anything you can only help them find it within themselves 8 Goals for this Course My goal is to help all students find within themselves sufficient knowledge of engineering analysis so that they will all get an A grade in the course What is your goal for this course What will you do to achieve that goal http space about com od astronomyhistory a galileoquotes htm 9 How to get your A How to Get your A Part II Spend six to ten hours per week outside class studying for the course Prepare for lecture and be ready to ask questions Read the assigned reading before class Download print and review the lecture presentations before class Use these as notes so that you can follow the lecture write additional notes on these presentations Study with fellow students and try to answer each other s questions Do the homework assignments Contact me by email telephone or office visits to ask questions Develop a good working relation with other members the class Participate in class discussions 11 ME 501B Engineering Analysis 10 12 2 Introduction Eigenfunction Expansions January 21 2009 What I will do to help Preliminary Assessment Designed to help instruction One set of questions on student background Second set of questions is ungraded quiz Take about 10 minutes for assessment Hand yours in when finished Arrive at class a few minutes early to answer any questions you may have Give lectures that stress application of basics to problem solving Return homework and exams promptly so that you can learn from your errors Be available for questions in my office visit or telephone or email Will call time when most students are done Send entire class emails as appropriate 13 Kinds of Problems 14 Energy Conservation Example General case of a flux proportional to a gradient of some potential Net heat input qx qx x y z cp x y z T t Here dQ cpdT Divide by x y z and take x 0 limit q x q x x q T x c p x x t x 0 Heat transfer qx k T x Mass diffusion jx D x Also applicable to simplified analyses of stress velocity and currents Lim Substitute qx k T x Conservation equation relates flow which is flux times area to conserved quantity Q Net inflow accumulation rate x y z Q t q T T T x k k x x x x t x Multidimensional problems have terms in y and z directions like the x direction term c p 15 Energy Example Continued c p Specific Examples T T T T k k k k T t x x y y z z One two and three dimensional transient Cartesian problems 2T 2T T 2 2 x t y 2 2 2 T T T T 2 2 2 x t y z For constant k c p 2T T 2 t x T T T T k 2 2 2 k 2T x t y z 2 2 2 Del operator in Cartesian coordinates i j k x y z T c p k 2T t 2 2 x k c p 2 2 y 2 Steady problems Laplace s equation 2 z 2 d 2T T 2T t dx 17 ME 501B Engineering Analysis 16 0 2T 2T y 2 x 2 2 2 T T T 0 x 2 y 2 z 2 2 2 0 18 3 Introduction Eigenfunction Expansions January 21 2009 Vector Spaces Orthogonal Functions Abstract concept that generalizes usual vectors from mechanics Define linear independence Basis is a complete set of linearly independent vectors which can be used to express any vector in the space Vector dot product a b generalized to inner product a b Functions such as sin n x L form a vector space in the region 0 x L The inner product defined below shows that this is a set of orthogonal functions L 1 n m L n x m x f n f m sin sin dx nm nm 2 L L 0 n m 0 The set of functions at g 2 sin n x L L the right is orthonormal n a and b orthogonal if a b 0 For functions f g fgdx 19 The Sturm Liouville Problem 20 The Sturm Liouville Problem II Why do we care about this problem Solution of partial differential equations PDE based on separation of variables Obtain solutions to PDEs as product of ordinary differential equation ODE solutions Want to obtain solution for general initial or boundary conditions e g u x t 0 f x Will show that