Differential Equations Brannan Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 09: Partial Differential Equations and Fourier Series

Chapter 9 Chapter 9 Partial Differential Equations and Fourier Series In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial, rather than ordinary, differential equations. We study a method known as separation of variables to solve partial diff. eqs. We discuss Fourier series. Illustrate solving method with examples of heat conduction, wave propagation, and potential theory.

Chapter 9 - Chapter 9 - Partial Differential Equations and Fourier Series 9.1 Two-Point Boundary Value Problems 9.2 Fourier Series 9.3 The Fourier Convergence Theorem 9.4 Even and Odd Functions 9.5 Separation of Variables; Heat Conduction in a Rod 9.6 Other Heat Conduction Problems 9.7 The Wave Equation: Vibrations of an Elastic String 9.8 Laplace’s Equation

9.1 Two-Point Boundary Value Problems Boundary Conditions - The value of the y or its derivative is specified at two different points. Recall that initial conditions specify the value of y and y' at the same point. A differential equation and suitable boundary conditions form a two-point boundary value problem. A typical example is the differential equation y'' + p(x)y' + q(x)y = g(x) with the boundary conditions y(α) = y 0, y(β) = y 1. If g(x)=0, y 0 = y 1 =0, problem is called homogeneous. Otherwise, the problem is nonhomogeneous.

Example Solve the boundary value problem y'' + 2y = 0, y(0) = 1, y(π) = 0. Answer The general solution of the differential equation is y = c 1 cos√2 x + c 2 sin√2 x. Use boundary conditions given to find c 1 and c 2. Thus the solution of the boundary value problem is y = cos√2 x − cot√2 π sin√2 x.

Eigenvalue Problems Consider the problem consisting of the differential equation y'' + λy = 0, together with the boundary conditions y(0) = 0, y(π) = 0. The values of λ for which nontrivial solutions (y≠0) to the diff. eq. occur are called eigenvalues,and the nontrivial solutions themselves are called eigenfunctions.

Eigenvalue Problems (Ctd) The boundary value problem y'' + λy = 0,y(0) = 0, y(π) = 0 has an infinite sequence of positive eigenvalues λ n = n 2 for n = 1, 2, 3,... and that the corresponding eigenfunctions are proportional to sinnx. Further there are no other real eigenvalues.

Eigenvalue Problems (Ctd.) The boundary value problem y'' + λy = 0,y(0) = 0, y(π) = 0 has an infinite sequence of positive eigenvalues and corresponding eigenfunctions given by λ n = n 2 π 2 /L 2, y n (x)= sin(nπx/L) for n = 1,2,3,... Further there are no other real eigenvalues.

9.2 Fourier Series If you can express a given function as an infinite sum of sines and/or cosines that is called the Fourier series of the function. The Euler–Fourier Formulas

Periodicity of the Sine and Cosine Functions A function f is said to be periodic with period T > 0 if the domain of f contains x + T whenever it contains x, and if f (x + T ) = f (x). The smallest such value of T is called the fundamental period of f. Example: The period T of sin(mπx/L) and cos(mπx/L) is given by T = 2πL/mπ = 2L/m.

Orthogonality of the Sine and Cosine Functions The standard inner product (u, v) of two real- valued functions u and v on the interval α ≤ x ≤ β is defined by The functions u and v are said to be orthogonal on α ≤ x ≤ β if their inner product is zero. A set of functions is said to be mutually orthogonal if each distinct pair of functions in the set is orthogonal. Example: The functions sin(mπx/L) and cos(mπx/L), m = 1, 2,... form a mutually orthogonal set of functions on the interval −L ≤ x ≤ L.

Example This function represents a triangular wave With the Fourier series for f given by

Example Let f (x) ={ 0, −3 < x < −1, {1, −1 < x < 1, {0, 1 < x < 3 and suppose that f (x + 6) = f (x); Find the coefficients in the Fourier series for f. Answer: The Fourier series for f is

9.3 The Fourier Convergence Theorem A function f is said to be piecewise continuous on an interval a ≤ x ≤ b if the interval can be partitioned by a finite number of points a = x 0 < x 1 < …< x n = b so that 1. f is continuous on each open subinterval x i−1 < x < x i. 2. f approaches a finite limit as the endpoints of each subinterval are approached from within the subinterval.

THEOREM Suppose that f and f' are piecewise continuous on the interval −L ≤ x < L. Further suppose that f is defined outside the interval −L ≤ x < L so that it is periodic with period 2L. Then f has a Fourier series whose coefficients are given by Eqs. The Fourier series converges to f (x) at all points where f is continuous, and to [ f (x+) + f (x−)]/2 at all points where f is discontinuous.

Example Let f (x) = { 0, −L < x < 0, { L, 0 < x < L, and let f be defined outside this interval so that f (x + 2L) = f (x) for all x. Find the Fourier series for this function and determine where it converges. Answer

Gibbs phenomenon Fourier series at points of discontinuity do not converge smoothly to the mean value. Instead, they tend to overshoot the mark at each end of the jump, as though they cannot quite accommodate themselves to the sharp turn required at this point. See the Fourier series for the square wave in fig.

9.4 Even and Odd Functions f is an even function if its domain contains the point −x whenever it contains the point x, and if f (−x) = f (x) for each x in the domain of f. f is an odd function if its domain contains −x whenever it contains x, and if f (−x) = −f (x)

5 Elementary properties of even and odd functions 1. The sum (difference) and product (quotient) of two even functions are even. 2. The sum (difference) of two odd functions is odd; the product (quotient) of two odd functions is even. 3. The sum (difference) of an odd function and an even function is neither even nor odd; the product (quotient) of two such functions is odd. 4. If f is an even function, then 5. If f is an odd function, then

Cosine Series - the Fourier series of any even function Suppose that f and f are piecewise continuous on −L ≤ x < L and that f is an even periodic function with period 2L. Then f (x) cos (nπx/L) is even and f (x) sin (nπx/L) is odd. The Fourier coefficients of f are then given by Thus f has the Fourier series

Sine Series - the Fourier series for any odd function Suppose that f and f are piecewise continuous on−L ≤ x < L and that f is an odd periodic function of period 2L. Then it follows from properties 2 and 3 that f (x) cos (nπx/L) is odd and f (x) sin (nπx/L) is even. In this case the Fourier coefficients of f are and the Fourier series for f is of the form

Example Let f (x) = x, −L < x < L, and let f (−L) = f (L) = 0. Let f be defined elsewhere so that it is periodic of period 2L (see Figure). This function is called a sawtooth wave. Find the Fourier series for this function. Answer:

9.5 Separation of Variables; Heat Conduction in a Rod The Heat Equation The variation of temperature in the bar is governed by a partial differential equation called the heat Conduction equation and has the form α 2 u xx = u t, 0 0, (1) where α2 is a constant known as the thermal diffusivity. The parameter α2 depends only on the material from which the bar is made and is defined by α 2 = κ/ρs, (2) where κ is the thermal conductivity, ρ is the density, and s is the specific heat of the material in the bar. The units of α 2 are (length)2/time.

Initial & Boundary Conditions Initial Condition. In addition to (1), the initial temperature distribution in the bar is given; thus u(x, 0) = f (x), 0 ≤ x ≤ L, (3) where f is a given function. Boundary Conditions. If the ends of the bar are held at fixed temperatures: the temperature T1 at x = 0 and the temperature T2 at x = L. In this section we will assume that u is always zero when x = 0 or x = L: u(0, t) = 0, u(L, t) = 0, t > 0. (4) The fundamental problem of heat conduction is to find u(x, t) that satisfies the differential equation (1) for 0 0, the initial condition (3) when t = 0, and the boundary conditions (4) at x = 0 and x = L.

The Method of Separation of Variables Assume u(x, t) is a product of two functions, one depending only on x and the other depending only on t; thus u(x,t) = X(x)T(t). Substitute into (1) and obtain X''/X= 1/α 2 T'/ T = −λ. Hence the partial differential equation Reduces to two ordinary differential equations with X(0)=0 and X(L)=0.

The Method of Separation of Variables (Ctd.) The functions satisfy the partial differential equation (1) and the boundary conditions (4) for each positive integer value of n. The functions u n are sometimes called fundamental solutions of the heat conduction problem (1), (3), and (4).

The Method of Separation of Variables (Ctd.) The solution of the heat conduction problem of Eqs. (1), (3), and (4) is given by the series with the coefficients computed from

Example Find the temperature u(x, t) at any time in a metal rod 50 cm long, insulated on the sides, which initially has a uniform temperature of 20◦C throughout and whose ends are maintained at 0◦C for all t > 0. Answer The temperature in the rod satisfies the heat conduction problem (1), (3), (4) with L = 50 and f (x) = 20 for 0 < x < 50. Substitute and simplify to obtain