Classification multidimensional PDEs March 2 2009 Overview Classification of PDEs and Multidimensional PDEs Review last class Wave equation solutions by separation of variables and D Alembert approach Characteristics and classification of partial differential equations Larry Caretto Mechanical Engineering 501B General analysis Parabolic equations Elliptic equations Hyperbolic equations Seminar in Engineering Analysis March 2 2009 Solving a wave equation problem 2 Midterm Exam Review Wave Equation Wednesday March 11 Covers material on diffusion and Laplace equations Includes material up to and including tonight s February 23 lecture and homework for Monday March 2 Open book and notes including homework solutions Focus on working with existing solutions Usual assumption u x t X x T t Result is 1 1 2T t 1 2 X x 2 function of t X x x 2 c 2 T t t 2 equal to u x t T t X x function of x A sin ct B cos ct C sin x D cos x Use above solution as starting point Boundary conditions at x 0 and x L Initial conditions on u and u t at x 0 3 Review General Solution u 2 Solution for u x t with initial and boundary conditions u x 0 f x u x 0 g x u 0 t u L t 0 c2 Review Use of Trig Identities u 2 2t x 2 0 x L t 0 c is wave speed L 2 m x g x sin dx m L 2 m x f x sin dx L0 L L Bm 0 5 ME 501B Engineering Analysis sin y cos y sin z n ct n x n ct u x t An sin sin Bn cos L L L n 1 n ct n x n ct u x t An sin sin Bn cos L L L n 1 Am 4 2 sin z cos y sin z y sin z y 2 sin z sin y cos z y cos z y u x t 1 2 An cos n 1 n x ct n x ct cos L L n x ct n x ct Bn sin sin L L 6 1 Classification multidimensional PDEs March 2 2009 Review D Alembert Solution Review D Alambert II Wave phenomena u x t is wave amplitude varying with 2 2 u space x and time t 2 u c c is wave speed 2t x 2 Over any x region and t 0 We see that the solution obtained by separation of variables agrees with the D Alambert solution for one case Solution shows propagation of wave shapes without damping D Alembert solution using x ct and x ct u x 0 f x and u x t 0 g x Look at meaning of f x ct and f x ct As time increases f x ct retains the shape of the initial condition and moves to the left Similarly f x ct retains its shape and moves to the right x ct u x t 1 f x ct f x ct 1 g d 2 2c x ct 7 8 Review Wave Propagation Review Boundaries With g x 0 solution is Fourier sine series which is periodic odd function 0 1 5 1 0 5 0 Periodic extensions 3 2 1 0 1 2 3 4 n x sin L Actual solution 0 5 2 4 n 1 n 0 5 1 5 B 1 f x time t t 0 ct 1 ct 2 ct 3 x ct x ct x ct x ct u x 0 f x Bm 5 distance x 9 20 m 2 2 0 5 1 1 5 2 x m 2m 3m 2 sin 2 sin 5 sin 5 10 1 f x ct Review Time Evolution Look at evolution f x ct f x ct u x t when ct 0 4 2 f x ct 2 f x 0 4 2 f x ct 2 f x 0 4 2 f x 0 5 ct 0 450 0 0 5 0 0 2 0 4 0 6 0 8 1 0 8 1 0 8 1 x 1 f x ct 1 0 5 0 0 5 ct 0 450 0 0 5 0 5 0 0 2 0 4 Initial Condition 1 0 6 x 1 1 5 1 0 5 0 x 0 5 1 For larger values of x ct periodic extensions move into 0 x L 1 1 5 2 u x t 2 0 5 ct 0 450 0 0 5 11 0 0 2 0 4 0 6 x ME 501B Engineering Analysis Phase behavior of sine function causes initial wave form to be reflected at boundaries 2 Classification multidimensional PDEs March 2 2009 Review Characteristics Review Classification of PDEs Used to classify second order PDEs The general second order PDE in two variables is classified as follows 2u 2u 2u u u B C 2 D x y u 0 2 x x y y x y dy B B 2 4 AC dx 2A If B2 4AC 0 the PDE is called elliptic and has no real characteristic directions If B2 4AC 0 the PDE is parabolic and has one repeated characteristic direction If B2 4AC 0 the PDE is hyperbolic and has two real characteristic directions Characteristics slopes gives region of influence and domain of dependence dy B B 2 4 AC dx 2A A Analysis gives slope of characteristics 13 A 14 u u 2u 2u 2u C 2 D x y u 0 B 2 x y y x y x 2u 2u D 0 x 2 y 2 Wave Equation Characteristics Laplace Poisson Helmholtz equations A C 1 B 0 B2 4AC 0 are elliptic no real characteristics Compute characteristic directions Wave equation A 1 B 0 2u 2u c2 2 0 2 C c2 B2 4AC c2 0 x t Diffusion equation A 2u 2 D 0 0 B C 0 B2 4AC x 0 is parabolic one characteristic Wave equation B2 4AC 0 is hyperbolic two real characteristics 15 A dx B B 2 4 AC 0 0 2 4 1 c 2 c dt 2A 2 1 16 Wave Equation Characteristics II Review Wave Propagation Compute characteristic directions with order of variables reversed 2u x 2 A 2u x 2 2u t 2 B 0 Characteristic slope dt dx 1 c Characteristic slope dx dt 1 c Wave equation A c2 B 0 C 1 B2 4AC c2 0 2u u u 2u C 2 D t x u 0 x t x t t dt B B 2 4 AC 0 0 2 4 1 c 2 1 dx 2A c 2 c2 5 4 3 2 1 0 1 2 3 4 5 distance x 17 ME 501B Engineering Analysis t 0 ct 1 ct 2 ct 3 x ct x ct x ct x ct time t c2 u u 2u 2u 2u D t x u 0 C B 2 2 x t x t x t 18 3 Classification multidimensional PDEs March 2 2009 Behavior of Equation …