Midterm Review March 9 2009 Overview Midterm Review Review last class General approach Diffusion equation in two space dimensions Three dimensional Laplace equation Larry Caretto Mechanical Engineering 501B Review for midterm Seminar in Engineering Analysis March 9 2009 Sturm Liouville solution eigenvalues General approach for PDEs Transformations and superposition Diffusion equation Laplace equation Rectangular and cylindrical coordinates Review 2D Diffusion Review f x y U a Constant Two dimensional diffusion 1 u 2u 2u equation for u x y t t x 2 y 2 t 0 0 x L 0 y H u x y 0 f x y u 0 y t u L y t u x 0 t u x H t 0 u x y t Cnm e n 2 m 2 t L H n 1 m 1 Cnm 4 HL HL n x m y sin dxdy L H f x y sin 00 H L 16 4 n x m y U sin sin dxdy mn 2 HL 0 0 L H 0 n 2n 1 Cnm n x m y sin sin L H 4 n x m y f x y sin sin dxdy HL 0 0 L H H L Cnm m 2m 1 u x y t 16U 2 2 n2 m2 t L H e n 0 m 0 n m odd m and n otherwise y x sin n sin m H L 3 4 Review Nonzero Boundaries Review 3D Laplace Sturm Liouville eigenfunction expansions require zero boundary conditions For nonzero boundaries split solution as in 1D case u x y t v x y t w x y v satisfies diffusion equation with zero boundary conditions w satisfies Laplace s and diffusion equation with nonzero boundary conditions u satisfies diffusion equation with u x y t w x y at boundaries 5 ME 501B Engineering Analysis 2 Use combination of separation of variables and superposition Start with basic solution that has homogenous boundary conditions at five surfaces and u x y W uW x y 2u 2u 2 2u 0 0 x L 0 y H 0 z W x y z 2 u 0 y z u L y z u x 0 z u x H z u x y 0 0 2 6 1 Midterm Review March 9 2009 Review 3D Laplace II Midterm Exam Solution of Laplace equation in x y z similar to diffusion solution in x y t Have hyperbolic cosine in z direction instead of exponential time decay u x y z n 1 m 1 Cnm n z m z n x m y sin sin L H L H Cnm sinh 4 n W m W HL sinh H L HL n x m y sin dxdy L H uW x y sin 00 Wednesday March 11 Covers material on diffusion and Laplace equations 2 independent variables only Includes material up to and including homework for Monday March 2 Open book and notes including homework solutions and integral tables Typical possible integrals sin2ax xsin ax 7 Sturm Liouville d dy r x dx dx q x p x y 0 k1 y a k 2 l 1 y b l 2 Partial Differential Equations dy dx x a dy dx x b Solving PDEs 0 Basic idea is to get solution to PDE as sum of eigenfunctions than can be used to represent an initial or boundary condition 0 Can expand any f x in terms of complete set of eigenfunctions ym b f x a m y m x m 0 y f am m ym ym p x y m x f x dx a b p x y a m x ym x dx Ability to get such a set of eigenfunctions assured if we have a Sturm Liouville problem Key element of such a problem is homogenous differential equation and boundary conditions Will use various transforms to get this problem For diffusion equation use u x t v x t w x For Laplace s equation use superposition 9 PDE Boundaries 10 Solving PDE Start Boundary conditions Fixed value first kind or Dirichlet e g u 0 at x 0 Fixed gradient second kind or Newmann e g u x 0 at x L Mixed third kind a u y y H b uy H 0 Use transforms like u v w diffusion or superposition Laplace if boundary conditions do not equal zero Problems in cylinder and sphere require u to be finite at r 0 11 ME 501B Engineering Analysis 8 Perform necessary operations if boundaries not homogenous Diffusion define u x t v x t w x v satisfies diffusion equation with zero boundary conditions w satisfies boundary Can also uses this for source term Laplace equation use superposition Solution is sum of two or more solutions each of which has only one nonzero boundary Each solution has consistent boundary condition kind for zero and nonzero parts 12 2 Midterm Review March 9 2009 Separation of Variables Separation of Variables Result Not needed if existing separation of variables solution is available Starting solutions Diffusion equation rectangular geometry Set variable in PDE u as product of two functions of one variable u F x1 G x2 Substiute product into PDE and differentiate Divide by original product solution to get two terms one with F only and one with G Set one term to a 2 to get eigenfunctions Solve resulting pair of ODEs u x t e 2 t C1 sin x C2 cos x Diffusion equation cylindrical geometry u r t e 2 t C1J 0 r C2Y0 r Diffusion equation spherical geometry 2 cos r sin r C2 u r t e t C1 r r 13 Separation of Variables Result II Rectangular Laplace starting solutions Rectangular nonzero BC at y 0 or y H u x y A sin x B cos x C sinh y D cosh y Rectangular nonzero BC at x 0 or x L u x y A sinh x B cosh x C sin y D cos y Cylindrical nonzero BC at z 0 or z H u r z A sinh z B cosh z CJ 0 r DY0 r Cylindrical nonzero BC at r Ro or R Ri u r z A sin z B cos z CI 0 R DK 0 R 0 14 Fitting Boundary Conditions Start with homogenous boundary conditions variable gradient or sum 0 Eliminate some constants in starting solutions Constants will be zero or be related to each other e g C DI0 Ro K0 Ro One boundary condition will lead to eigenvalues and eigenfunctions Final result should be product solution with one eigenfunction and one constant 15 Eigenfunction Expansions Diffusion Equation For nonzero boundary conditions or additional source term S x a function of x only use u x t v x t w x Most general solution is sum of infinite series of eigenfunctions each with it s own constant nCnFn Gn Use nonzero boundary condition initial condition in diffusion equation to fit constants via eigenfunction expansion b Cn f a boundary condition b p Fn n d Gn B p Fn n d …