Introduction to PDE\'s PDF

Preview

Summary

Introduction to the various different classes of partial differential equations. ...

Description

Math 471 Notes

Jan 18, 17

1

Course Information 1. Classes MWF: 10:50 am - 12:50 am, T: 10:05 am - 11:30 am • Tuesdays primarily for exams unless Friday class gets cancelled. 2. Grading • Quizzes (Lectures and Quizzes) - 20 % • Midterms - 25 % x 2 • Final - 30 % – Problems in midterm, and final similar to Homework and problems done in class • Attendance helps for final letter grade

2

Introduction to PDE’s

ODE: u = u(x) (Single variable) PDE: u = u(x, t) (More independent variables) Definition: A PDE (Partial Differential Equation) is an equation that contains partial derivatives. Ex: ut = uxx, where, u = u(x, t) **Heat Equation** Notation: ∂u ∂u , ux = ut = ∂t ∂x ∂2u , uxx = ∂t2   ∂u ∂ = = ∂x ∂t

utt = uxt

1

∂2u ∂x2 ∂2u ∂x∂y

Ex: utt = uxx + uyy , where, u = u(x, y, t) Physical meaning of PDE’s    u : Temperature u(x, t) = x : Location   t : Time Heat equation in one-dimension.

3

A few well known PDE’s 1. ut = uxx (Heat Equation in 1-D) *Spatial Dimension 2. ut = uxx + uyy (Heat Equation in 2-D) 3. uxx + uyy = 0 (Laplace’s Equation) 4. urr + 1r ur + 1r uθθ = 0 (Laplace’s Equation in Polar Coordinates) 5. utt = uxx + uyy + uzz (Wave Equation in 3-D) *Spatial Dimensions 6. utt = uxx + αut + βu (Telegraph Equation)

4

Applications

Example Heat flow, mechanics. Wave motion, magnetism (Maxwell’s Equations)

5

Classifications

: 1. Order of P.D.E: Order of the highest partial derivative of the equation Example ut = uxx (Second Order PDE) *Heat Equation ut = ux (First Order PDE) ut = uuxxx + sin(x) (Third Order PDE) 2. Linearity: Linear Equation U and all of its derivatives appear in a linear fashion ut − uxx = 0(Linear)

(1)

x · ut + uxx = 0(Linear)

(2)

xux + yuy + u = 0(Linear )

(3)

u · ut − uxx = 0(N on − Linear)

(4)

2

6

General Form for Second Order Linear Equation in Two Variables Auxx + Buxy + C uyy + Dux + Ruy + F U = G

A,B,C,D,E,F → Coefficients (Constants or Variables) 1.

• When G = 0, Homogeneous Equation • When G 6= 0, Non-homogeneous Equation

2.

• A,B,C,D,E,F ← Constants, Constant Coefficients • A,B,C,D,E,F ← Functions of XY, Variable Coefficients

Example ut t = e−t ux x + sin(t) (Second Order, Linear, Non-Homogeneous) Example Heat Equation: uxx − ut = 0 (Second Order, Linear, Homogeneous)

7

Classifications of Second Order Linear Equation in Two Variables Auxx + Buxy + C uyy + Dux + Ruy + F U = G

(a) Parabolic Equation: B 2 − 4AC = 0 Example Heat Equation: ut = uxx, A = 1, B = 0, C = 0

*Used to study diffusion processes. (b) Hyperbolic Equation: B 2 − 4AC > 0 Example Wave Equaion in 1D: utt = uxx ,A = 1, B = 0, C = −1 *Used to study oscillations and vibrations (c) Elliptic Equation: B 2 − 4AC < 0 Example Laplace’s Equation: uxx = −uyy, A = 1, B = 0, C = 1 *Used to study steady-state phenomenon

3...