Partial Differential Equations Assoc. Prof. Dr. Siridech Boonsang Electrical Engineering Partial Differential Equations Background • Physical problems are governed by many PDEs • Some are governed by first order PDEs • Numerous problems are governed by second order PDEs • A few problems are governed by fourth-order PDEs. Examples 2 T T Heat Conduction Equation (1 D) : C 2 t x 2T 2T T Heat Conduction Equation ( 2 D) : C x 2 y 2 t 2 D Laplace Equation : 2 2 2 D Poisson Equation : 2 x 2 x 2 2 2 y 2 y 2 2 0 f ( x, y ) • Laplace’s Equation: 2u = uxx +uyy +uzz = 0 • unknown: u(x,y,z) • gravitational / electrostatic potential Heat Equation: ut = a22u unknown: u(t,x,y,z) heat conduction Wave Equation: utt = a22u unknown: u(t,x,y,z) wave propagation Examples (contd.) 1 D Wave Equation : 2 u t 2 C 2 u x 2 2 D Wave Equation ( vibrating membrane) : 2 2u 2u u C 2 x 2 y 2 t 4thOrder Equations : 2 4 u u C 0 2 4 t x 4 4 2 4u u u u D 2 h 0 2 2 4 2 x 4 x y y t ( Vibrating beam) ( Vibrating plate) • Schrödinger Wave Equation • quantum mechanics • (electron probability densities) • Navier-Stokes Equation • fluid flow (fluid velocity & pressure) Classification of Partial Differential Equations (PDEs) There are 6 basic classifications: (1) Order of PDE (2) Number of independent variables (3) Linearity (4) Homogeneity (5) Types of coefficients (6) Canonical forms for 2nd order PDEs (1) Order of PDEs The order of a PDE is the order of the highest partial derivative in the equation. Examples: u 2 u 2 t x (2nd order) u u (1st order) t x u 3u u 3 sin x (3rd order) t x (2) Number of Independent Variables Examples: (2 variables: x and t) (3 variables: r, q, and t) u 2 u 2 t x u 2 u 1 u 1 2 u 2 2 2 t r r r r q (3) Linearity PDEs can be linear or non-linear. A PDE is linear if the dependent variable and all its derivatives appear in a linear fashion (i.e. they are not multiplied together or squared for example. 2u Examples: (Linear) t (Non-linear) u (Linear) (Non-linear) (Non-linear) (Linear) 2u u 0 t 2u 2u x 2 x 2 y y 2 e 2u x 2 sin t 0 2 u u x y u2 0 x y 2 u u y sin u e 2 x x 2u 2u 2u 2 2 sin x 2 x y y x 2 (Non-linear) 2 t u u 1 u y x (4) Homogeneity A PDE is called homogenous if after writing the terms in order, the right hand side is zero. Examples: 2u 2u 2 f ( x, y ) (Non-homogeneous) 2 x (Homogeneous) (Homogenous) y 2u u 0 2 t x 2u t 2 2u x 2 u Examples (Non-homogeneous) (Homogeneous) u u u5 x t ( u 5) ( u 5) u5 x t (5) Types of Coefficients If the coefficients in front of each term involving the dependent variable and its derivatives are independent of the variables (dependent or independent), then that PDE is one with constant coefficients. Examples (Variable coefficients) 2u x 2 (C constant; constant coefficients) x 2 u 2 y 2u x 2 2 0 C 2u t 2 0 (6) Canonical forms for 2nd order PDEs (Linear) 2u 2u 2u u u A 2 B C 2 D E Fu G x y x y x y (Standard Form) where A, B, C, D, E, F, and G are either real constants or real-valued functions of x and/or y. B 2 4AC 0 PDE is Elliptic B 2 4AC 0 PDE is Parabolic B 2 4AC 0 PDE is Hyperbolic Parabolic PDE solution “propagates” or diffuses Hyperbolic PDE solution propagates as a wave Elliptic PDE equilibrium This terminology of elliptic, parabolic, and hyperbolic, reflect the analogy between the standard form for the linear, 2nd order PDE and conic sections encountered in analytical geometry: Ax 2 Bxy Cy 2 Dx Ey F 0 2 B 4AC 0 one obtains the equation for an ellipse, for which when when B 2 4AC 0 one obtains the equation for a parabola, and when B 2 4AC 0 one gets the equation for a hyperbola. Examples 2u 2 2u 0 x y (a) Here, A=1, B=0, C=2, D=E=F=G=0 B24AC = 0 - 4(1)(2) = -8 < 0 this equation is elliptic. 2 2u 2 2 2u 0 x y (b) Here, A=1, B=0, C=-2, D=E=F=G=0 B2-4AC = 0 - 4(1)(-2) = 8 > 0 this equation is hyperbolic. 2 2 2u u 2 0 2 y x (c) Here, A=1, B=0, E=-2, C=D=F=G=0 B2-4AC = 0 - 4(1)(0) = 0 this equation is parabolic. Examples 2u 2u 2u 4 2 0 2 x y y x (d) Here, A=1, B=-4, C=1, D=E=F=G=0 B2-4AC = 16 - 4(1)(1) = 12 > 0 this equation is hyperbolic. 2u 2u 2u 3 2 4 5 2 0 x y x y (e) Here, A=3, B=-4, C=-5, D=E=F=G=0 B2-4AC = 16 - 4(3)(-5) = 76 > 0 this equation is hyperbolic. 2u 2u 2u u u 3 2 4 5 2 8 9 6u 27e xy x y x y y (f) x Here, A=3, B=-4, C=-5, D=8, E=- 9, F=6, G=27exy B2-4AC = 16 - 4(3)(-5) > 0 this equation is hyperbolic. Examples y 2u 2u 0 x y (g) Here, A=y, B=0, C=-1, D=E=F=G=0 B2-4AC = 0 - 4(y)(-1) = 4y for y>0, this equation is hyperbolic; for y=0, this equation is parabolic; for y<0, this equation is elliptic. 2 2 y Hyperbolic x Elliptic Parabolic Examples 2u 2 2u 2 u 2x (1 y ) 2 0 2 x y x y (h) Here, A=1, B=2x, C=1-y2, D=E=F=G=0 B2-4AC = 4x2 - 4(1)(1-y2) = 4x2+4y2-4 or x2+y2 >,=,< 0 y Hyperbolic Elliptic x Parabolic on surface of circle Examples 2u 2u u u y x y u0 2 x y x y x (i) Here, A=1, B=-y, C=0, D=E=F=G=0 B2-4AC = y2 for y=0, this equation is parabolic; for y0, this equation is hyperbolic. y Hyperbolic x Hyperbolic Parabolic Example (j) 2u 2u 2 (sin x ) 2 (sin 2x ) (cos x ) 2 x x y x y 2 2u Here, A=sin2x, B=sin2x, C=cos2x, D=E=F=G=0 B2-4AC = sin22x-4sin2xcos2x = 4sin2xcos2x-4sin2xcos2x = 0 this equation is parabolic everywhere. Solution Technique • Elliptic equations in engineering are typically used to characterize steady-state, boundary value problems. • For numerical solution of elliptic PDEs, the PDE is transformed into an algebraic difference equation. • Because of its simplicity and general relevance to most areas of engineering, we will use a heated plate as an example for solving elliptic PDEs. The Laplacian Difference Equations/ 2T 2T 2 0 2 Laplace Equation x y 2T Ti 1, j 2Ti , j Ti 1, j O[(x)2] x 2 x 2 2T Ti , j 1 2Ti , j Ti , j 1 O[(y)2] 2 2 y y Ti 1, j 2Ti , j Ti 1, j Ti , j 1 2Ti , j Ti , j 1 0 2 2 x y x y Ti 1, j Ti 1, j Ti , j 1 Ti , j 1 4Ti , j 0 Laplacian difference equation. Holds for all interior points • In addition, boundary conditions along the edges must be specified to obtain a unique solution. • The simplest case is where the temperature at the boundary is set at a fixed value, Dirichlet boundary condition. • A balance for node (1,1) is: T21 T01 T12 T10 4T11 0 T01 75 T10 0 4T11 T12 T21 0 • Similar equations can be developed for other interior points to result a set of simultaneous equations. • The result is a set of nine simultaneous equations with nine unknowns: 4T11 T21 T12 T11 4T21 T13 T21 75 T22 4T31 T11 0 T32 4T12 T22 T21 T12 T31 50 T13 4T22 T32 T22 75 T23 4T32 T12 T22 T32 0 T33 50 4T13 T23 175 T13 100 4T23 T33 T23 4T33 150