Introduction/Eigenfunction Expansions January 21, 2009ME 501B – Engineering Analysis 1Course Introduction and Course Introduction and Eigenfunction ExpansionsEigenfunction ExpansionsLarry CarettoMechanical Engineering 501BSeminar in Engineering Seminar in Engineering AnalysisAnalysisJanuary 21, 20092Overview• Course outline, schedule, grading, exams, homework, office hours, etc.• Problems considered• Eigenfunction expansions– Why do we care about this?– Sturm-Liouville problem– Complete orthogonal eigenfunctions– Examples of eigenfunction expansions• Sines and cosines• Bessel functions• Other eigenfunctions3Course Structure• ME 501AB is a one-year course in engineering and numerical analysis– 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 skills4Course Materials• 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 additions5Instructor and Course Data• [email protected] 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 outline6Course 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 functionsIntroduction/Eigenfunction Expansions January 21, 2009ME 501B – Engineering Analysis 27Course objectives II• 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 approaches8Course objectives III• 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 approachesYou cannot teach people anything; you can only help them find it within themselves.Galileo Galilei(1564-1642)http://space.about.com/od/astronomyhistory/a/galileoquotes.htm9 1010Goals 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?1111How to get your A• 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 1212How to Get your A, Part II• 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 discussionsIntroduction/Eigenfunction Expansions January 21, 2009ME 501B – Engineering Analysis 31313What I will do to help• 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– Send entire class emails as appropriate1414Preliminary 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– Will call time when most students are done15Kinds of Problems• General case of a flux proportional to a gradient of some potential– Heat transfer: qx= –k∂T/∂x– Mass diffusion: jx= –ρD ∂ω/∂x– Also applicable to simplified analyses of stress, velocity, and currents• Conservation equation relates flow, which is flux times area to conserved quantity, Q: Net inflow = accumulation rate = ΔxΔyΔz ∂Q/∂t16Energy Conservation Example• Net heat input, (qx–qx+Δx)ΔyΔz= ρcpΔxΔyΔz ∂T/∂t (Here dQ = ρcpdT)– Divide by ΔxΔyΔz and take Δx → 0limit– Substitute qx= –k∂T/∂x– Multidimensional problems have terms in y and z directions like the x direction termtTcxqqxqpxxxxxLim∂∂ρ=Δ−=∂∂−Δ+→Δ 0xTkxxTkxxqtTcxp∂∂∂∂=⎟⎠⎞⎜⎝⎛∂∂−∂∂−=∂∂−=∂∂ρ17Energy Example ContinuedTkzTkzyTkyxTkxtTcp∇•∇=∂∂∂∂+∂∂∂∂+∂∂∂∂=∂∂ρ– For constant k– “Del” operator in Cartesian coordinatesTkzTyTxTktTcp2222222∇=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∂∂ρ2222222zyxzyx∂∂+∂∂+∂∂=∇∂∂+∂∂+∂∂=∇ kjiTtTckTktTcpp22∇α=∂∂ρ=α∇=∂∂ρ18Specific Examples• One-, two- and three-dimensional transient, Cartesian problems• Steady problems (Laplace’s equation)⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂α=∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂α=∂∂∂∂α=∂∂222222222222zTyTxTtTyTxTtTxTtT000222222222222=∂∂+∂∂+∂∂=∂∂+∂∂=zTyTxTyTxTdxTdIntroduction/Eigenfunction Expansions January 21, 2009ME 501B – Engineering Analysis 419Vector Spaces• Abstract concept that generalizes usual vectors from mechanics• Define linear independence• Basis is a “complete set” of linearly independent “vectors” which can be