WMS. MA250 Introduction to Partial Differential Equations. Revision Guide. Written by Matthew Hutton and David McCormick

Transcription

1 WMS MA25 Introduction to Partia Differentia Equations Revision Guide Written by Matthew Hutton and David McCormick WMS

2 ii MA25 Introduction to Partia Differentia Equations Contents Introduction 1 1 First-Order Linear PDEs 1 2 The Wave Equation 3 3 The Diffusion Equation 4 4 Fourier Anaysis 6 5 The Lapace Operator 1 Introduction This revision guide for MA25 Introduction to Partia Differentia Equations has been designed as an aid to revision, not a substitute for it. PDEs is an appied course, in which the emphasis is on probemsoving; however, the course is rigorous as we. So, the best way to revise is to use this revision guide as a quick reference for the theory, and to just keep trying exampe sheets and mock exam questions. Hopefuy this guide shoud give you some confidence by showing you that there isn t too much to the modue: at east not as much materia as it appears from the wad of ecture notes and assignments you have amassed. Discaimer: Use at your own risk. No guarantee is made that this revision guide is accurate or compete, or that it wi improve your exam performance. Use of this guide wi increase entropy, contributing to the heat death of the universe. Contains no GM ingredients. Your mieage may vary. A your base are beong to us. Authors Written by Matthew Hutton (matthew.hutton@warwick.ac.uk) and David McCormick (d.s.mccormick@warwick.ac.uk) in 27. Based upon ectures given by Forian Thei at the University of Warwick in 27. History First Edition: May 23, 27.

3 MA25 Introduction to Partia Differentia Equations 1 Introduction Differentia equations, i.e. equations reating functions and their derivatives, are the foundation on which a of physics is buit; however, their abstract study has ed to many new advances in mathematics, not east the proof of the Poincaré conjecture. In MA133 Differentia Equations, we considered ordinary differentia equations, in which we ony had one independent variabe; these are in some sense one-dimensiona. But the word is not one-dimensiona: many physica probems depend on more than one independent variabe, and so when we differentiate we get partia derivatives in the mix. We are thus ed to study partia differentia equations. To save us a some writing, we denote partia derivatives using subscripts; so for a function u(x, y,...), we write u x := u x, u y := u y, u xx := 2 u x 2. Definitions. A partia differentia equation (abbreviated PDE) is an identity that reates the independent variabes, the dependent variabe u and its partia derivatives, i.e. an equation of the form F(x, y,..., u, u x, u y,...) =. (1) If F depends on x, y,...,u, u x, u y,... but not on the higher-order partia derivatives u xx, u xy, u yy,..., etc., then (1) is caed a first-order PDE. Simiary if F depends on x, y,...u, u x, u y,...,u xx, u xy,..., but not on higher-order derivatives, then (1) is caed a second-order PDE. A PDE is caed inear if F depends ineary on u, u x, u y,.... As with ODEs, inear PDEs are much easier to sove; e.g. u x + u y = is inear, but u x + uu y = is noninear. We ony consider first- and second-order inear PDEs in this course. In soving such PDEs, we wi make use of many resuts from MA131 Anaysis, MA244 Anaysis III, MA134 Geometry and Motion and MA231 Vector Anaysis; make sure you are famiiar with most of the major resuts such as directiona derivatives and the chain rue, the various generaisations of the Fundamenta Theorem of Cacuus (Green s theorem, the Divergence theorem, and Stokes theorem), and the various change of variabe formuae for integration. We wi aso ca upon soution methods for ODEs from MA133 Differentia Equations. Reca that D is the boundary of D and D := D D is the cosure of D. We use the notation u C 1 (D) to say that u: D R is continuousy differentiabe (i.e. C 1 ), and the domain can be extended to D. Uness otherwise stated, we wi assume that a derivatives exist and are continuous; this means that second derivatives commute, i.e. u xy = u yx. Furthermore, continuity of the partia derivatives aows us to differentiate under the integra sign: d dt b a f(x, t)dx = b a f (x, t)dx. t When finding a soution of an ODE of order m, we get m arbitrary constants, which can be determined by m initia conditions. When finding a soution of a PDE, we get arbitrary functions: for exampe, if u: R 2 R, the PDE u xx + u = ooks ike an ODE, but with an extra variabe y, so the soution is u = f(y) cos x + g(y) sin x, where f(y) and g(y) are two arbitrary functions of y. We need an auxiiary condition if you want to determine a unique soution; such conditions are usuay caed initia or boundary conditions. 1 First-Order Linear PDEs We start with some very simpe PDEs. Exampe 1.1. In some sense, the simpest possibe PDE is u x =, where u = u(x, y), which we can integrate to get u = f(y) as the genera soution (f(y) being some arbitrary function of y). Since the soutions don t depend on x, they are constant on the ines y = constant in the x y pane. Exampe 1.2. A sighty more compicated first-order equation is the transport equation; for some constant veocity c, the one-dimensiona transport equation is u t + cu x =. This describes transport phenomena such as a fuid moving in a pipe.

4 2 MA25 Introduction to Partia Differentia Equations 1.1 Method of Characteristics The most genera inear first-order PDE with constant coefficients is au x + bu y = (2) where a and b are constants, not both zero. We can sove this equation using the foowing two different methods. Geometric Method If u is a C 1 function, then au x + bu y is the directiona derivative of u in the direction (a, b). This directiona derivative must aways be zero, so u must be constant in the direction of (a, b). The ines parae to (a, b), caed characteristic ines, have bx ay = c for some constant c. The soution is constant on each such ine, thus u depends on bx ay ony, so for some arbitrary function f of one variabe, we have where f(x) = g(bx). ( u(x, y) = f (bx ay) = g x ay b Coordinate Method We can aso sove equation (2) agebraicay. First change coordinates to x = ax + by, y = bx ay. Then by the chain rue u x = au x + bu y and u y = bu x au y, hence au x + bu y = a(au x + bu y ) + b(bu x au y ) = (a 2 + b 2 )u x =. Then as a 2 + b 2, u x = with respect to the primed variabes. Hence ( u = f(y ) = f(bx ay) = g x ay ) b as before. 1.2 Variabe Coefficients We can generaise equation (2) to the case where a and b are not constants, but rather functions of x and y (though not of u). Exampe 1.3. Consider the PDE ) u x + yu y =. (3) Using the geometric method we deduce that the soutions are constant in the direction (1, y). Curves in the x y pane with (1, y) as tangent vectors have dy dx = y 1, which have the soutions y(x) = Cex for any C R. These curves are characteristic curves of equation (3). On each curve u is constant, since x u(x, Cex ) = u u + Cex x y = u x + yu y = for a functions u which satisfy equation (3). Thus u(x, Ce x ) = u(, Ce ) = u(, C) is independent of x, and depends ony on C Putting y = Ce x we find that C = e x y, so is the genera soution of (3). u(x, y) = f(e x y) This geometric method works nicey for any PDE of the form a(x, y)u x + b(x, y)u y =, since it reduces the task of finding the soution to soving the ODE dy b(x, y) = dx a(x, y). However, there is no genera soution for a first order PDEs, just as there is no genera soution formua for a first order ODEs.

5 MA25 Introduction to Partia Differentia Equations 3 2 The Wave Equation The wave equation is an important second-order PDE that describes the propagation of a variety of waves such as sound waves and eectromagnetic waves. For a constant c >, known as the wave speed, the one-dimensiona wave equation is u tt = c 2 u xx. Euer, Lagrange and Bernoui ooked at the probem of a vibrating string, such as in a musica instrument; we can interpret u(x, t) as the vertica dispacement of this vibrating string. We can formuate this equivaenty as ρu tt = Tu xx where ρ > is the mass density and T > is the eastic constant. (These two forms are equivaent if T ρ = c2.) 2.1 Soutions of the Unbounded Wave Equation If u soves the wave equation for a x, i.e. x is unbounded, then we can factor the derivative operators nicey: ( u tt c 2 u xx = t c ) ( x t + c ) u =. x Substituting ξ = x + ct and η = x ct reduces the wave equation to u ξη =. Integrating this with respect to ξ and then with respect to η gives that u = f(ξ) + g(η) for two arbitrary functions f and g, i.e. the genera soution is u(x, t) = f(x + ct) + g(x ct). 2.2 The Initia Vaue Probem In most cases we are interested in how the wave equation depends on the initia conditions u(x, ) = φ(x) and u t (x, ) = ψ(x), where φ and ψ are some given (though arbitrary) functions of x. Using u(x, t) = f(x + ct) + g(x ct), we can see that f(x) + g(x) = φ(x) = f (x) + g (x) = φ (x), cf (x) cg (x) = ψ(x) = f (x) g (x) = 1 c ψ(x). We wish to sove these to find the arbitrary functions f and g, so adding and subtracting these gives f = 1 (φ + 1c ) 2 ψ, g = 1 (φ 1c ) 2 ψ. Integration of these two equations then gives f(s) = 1 2 φ(s) + 1 2c s ψ( s)d s + A g(s) = 1 2 φ(s) 1 2c s ψ( s)d s + B where A and B are arbitrary constants. Since we know that f + g = φ, we can see that A + B =. Substituting s = x + ct in the formua for f and s = x ct in the formua for g and adding eads to d Aembert s formua: u(x, t) = f(x + ct) + g(x ct) = 1 2 [ φ(x + ct) + φ(x ct) + 1 c x+ct x ct ] ψ(s) ds. Exampe 2.1 (Standing Wave equation). For φ = and ψ = cos x, the soution of the wave equation is: u(x, t) = 1 c cosxsin(ct). This is known as a standing wave.

6 4 MA25 Introduction to Partia Differentia Equations 2.3 Causaity and Energy Conservation Remark 2.2 (Causaity). D Aembert s formua shows that the soution depends on the initia position and veocity at x ony for x [x ct, x + ct]. This is known as causaity, and is shown in more detai in figure 1; it means that no information can trave faster than the wavespeed c. t Domain of infuence x = x ct x = x + ct x x Domain of dependence Figure 1: Domain of dependence and infuence Another of the most basic facts about the wave equation is the principe of conservation of energy: Theorem 2.3 (Energy Conservation). Let u be a soution of the wave equation ρu tt = Tu xx such that u C 2 (R R), u(x, ) = φ(x), u t (x, ) = ψ(x) and for some R φ(x) = ψ(x) = if x > R. Then the tota energy E(t) := 1 ( ρ(ut (x, t)) 2 + T(u x (x, t)) 2) dx. 2 is constant with respect to time, i.e. E(t) = E() = 1 2 ( ρ ψ(x) 2 + T φ x (x) 2) dx. This can be proved easiy by differentiating E(t) with respect to time. 3 The Diffusion Equation Another important second-order PDE is the diffusion equation, which modes how something (ike heat) spreads through an object 1. For a constant k >, known as the diffusion coefficient, then the onedimensiona diffusion equation is u t = ku xx 3.1 Maxima, Uniqueness and Stabiity Diffusions are very different from waves, and this is refected in the mathematica properties of the equation. The soutions of the diffusion equation are harder to obtain, so we first find some genera properties of soutions. Theorem 3.1 (The Maximum Principe). Let u C 2( [, ] [, T] ) be a soution of the diffusion equation. Then u assumes its maximum on the set {(x, t) [, ] [, T] t =, x =, or x = }. Note that by appying the Maximum Principe to u(x, t) we obtain the Minimum Principe. 1 Accurate mathematica modes of the diffusion equation are pretty much aways non-trivia. This means that to sove them in this course we have to work with a simpified system.

7 MA25 Introduction to Partia Differentia Equations 5 Proof. Let M be the maximum vaue of u(x, t) on the set {(x, t) [, ] [, T] t =, x =, or x = }; we want to show that u(x, t) M for a (x, t) [, ] [, T]. Fix ε > and et v(x, t) = u(x, t) + εx 2. Ceary v(x, t) M + ε 2 for t =, x = or x =. Furthermore, v t kv xx = u t k(u + εx 2 ) xx = u t ku xx 2εk = 2εk <. If v(x, t) assumes its maximum at an interior point (x, t ), then v t = and v xx at (x, t ), hence v t kv xx, which is a contradiction. If v(x, t) assumes its maximum for some < x < and t = T, then v x (x, T) = and v xx (x, T), but as v(x, T) is a maximum v(x, T) v(x, T h) and hence 1 v t (x, T) = im h h [v(x, T) v(x, T h)], and so v t kv xx, which is again a contradiction. So v(x, t) assumes its maximum ony when t =, x = or x =. As v(x, t) must assume a maximum somewhere in [, ] [, T], we have that v(x, t) M +ε 2 for a (x, t) [, ] [, T], and hence that u(x, t) M +ε( 2 x 2 ). As ε was arbitrary, we have that u(x, t) M for a (x, t) [, ] [, T], as required. An appication of the Maximum Principe shows that soutions of the diffusion equation are unique. Theorem 3.2. Let u C 2( [, ] [, T] ) be a soution of the diffusion equation (u t = ku xx ). If u satisfies the initia condition u(x, ) = φ(x) and the boundary conditions then u is unique. u(, t) = g(t), u(, t) = h(t) Proof. Let u (1), u (2) C 2( [, ] [, T] ) be two soutions of the diffusion equation which satisfy the initia boundary conditions with the same functions for φ, g and h. Then et v(x, t) = u (1) (x, t) u (2) (x, t). Since the diffusion equation is inear, v(x, t) is aso a soution of the diffusion equation. Furthermore, v(x, ) = v(, t) = v(, t) =. The Maximum Principe then impies that v(x, t), whie the Minimum Principe impies that v(x, t), hence v(x, t) = for a x and t. The above proof aso hods for the so-caed inhomogeneous heat equation u t = ku xx + f(x, t), since when we subtract two soutions of this equation the f(x, t) terms cance. (f is usuay known as the heat source.) The second fundamenta principe for the diffusion equation is stabiity. For the wave equation, we found that a certain integra, the energy, is a constant of the motion. For the diffusion equation, we can show that the foowing energy estimate hods: Theorem 3.3 (Stabiity). Let u (1), u (2) C 2( [, ] [, ) ) be two soutions of the inhomogeneous heat equation u t = ku xx + f(x, t). If u (1) and u (2) satisfy the initia conditions u (1) (x, ) = φ 1 (x), u (2) (x, ) = φ 2 (x), and the boundary conditions u (1) (, t) = u (2) (, t) = g(t), u (1) (, t) = u (2) (, t) = h(t), then 1 ( u (1) (x, t) u (2) (x, t) ) 1 2 ( dx φ1 (x) φ 2 (x) ) 2 dx. (4) The right-hand side of the inequaity (4) measures the nearness of the initia data, and the eft-hand side the nearness of the soutions at any ater time; hence the soutions are, in the square-integra sense, stabe; if we start nearby (at t = ), then we stay nearby. (The maximum principe aso proves stabiity, but in the uniform sense.)

8 6 MA25 Introduction to Partia Differentia Equations 3.2 Soutions of the Unbounded Diffusion Equation We now find soutions to the initia vaue probem u t = ku xx, u(x, ) = φ(x) (5) for some u C 2( R [, ) ), i.e. when x is unbounded. This is harder and done very differenty than previous methods. In so doing, we use five basic invariance properties of the diffusion equation: Proposition 3.4 (Properties of the diffusion equation). If u C 2( R [, ) ) satisfies the diffusion equation, then the foowing aso satisfy the diffusion equation: 1. The transate u(x y, t), for any fixed y. 2. Any derivative u x, u t, u xx etc.. 3. Any inear combination of soutions, e.g. c 1 u + c 2 v (if v aso soves the diffusion equation). 4. An integra of soutions such as x u(s, t)ds. 5. The diated function f a (x) = u( ax, at) for any a >. Using this, it can be shown that the unique soution of (5) is u(x, t) = 1 4πkt e (x y)2 /4kt φ(y)dy provided that φ decays fast enough to ensure that the integra exists. 3.3 Comparison of the Wave and Diffusion Equations Having ooked at both the wave and diffusion equations we compare their various properties: Property Diffusion Wave Speed of propagation Infinite Finite ( c) Singuarities at t > Immediatey ost Trave aong characteristic ines Soutions exist for t > YES YES Soutions exist for t < NO YES Maximum Principe YES NO Information ost YES, (gradua) oss of information NO, transported 4 Fourier Anaysis Fourier series are an important way of finding soutions of PDEs. In order to motivate their study, we first consider the more physicay reaistic case of bounded intervas, rather than infinite ones as studied previousy. 4.1 Boundary Conditions and Separation of Variabes When we study the wave and diffusion equations on finite intervas, we must specify the vaues of u at the boundary of the interva in question in order to find a particuar soution for a physica probem. There are two common ways of doing this: Dirichet boundary conditions: Let u C([, ] [, T]). Then Dirichet boundary conditions take the form u(, t) = a, u(, t) = b, for some a, b R. (Often we take a = b.) In terms of the wave equation, the Dirichet boundary conditions mode the assumption that the ends of a vibrating string are camped. In genera, if u C(D) for some open set D R k, then Dirichet boundary conditions take the form u(x) = for a x D.

9 MA25 Introduction to Partia Differentia Equations 7 Neumann boundary conditions: Let u C([, ] [, T]). Neumann boundary conditions take the form u x (, t) = a, u x (, t) = b, for some a, b R. For the wave equation, Neumann boundary conditions mode the assumption that we are puing with a constant force on the ends of a vibrating string. In genera, if u C 1 (D) for some open set D R k, then Neumann boundary conditions take the form u n (x) = for a x D, where u n (x) := u(x) n(x) = k D. i=1 u n i (x) n i (x) if n(x) R k is the outward norma vector of An important method of finding soutions is that of separation of variabes. In this method, we assume that a soution u(x, t) of a particuar PDE can be written as u(x, t) = X(x)T(t), where X and T are functions of one variabe. Often we suppose that one of the functions is can be expressed as a Fourier series; typicay such a separation Ansatz wi be given as a starting point, as finding such an Ansatz requires some ingenuity. Exampe 4.1. Consider again the wave equation with Dirichet boundary conditions: u tt = c 2 u xx, u(, t) = u(, t) =. (6) Now we fix some specia initia conditions, as originay done by Euer: ( ) kπx u(x, ) = a k sin, u t (x, ) = b k sin ( kπx ). (7) We assume that the soution can be written as u(x, t) = v(t)sin ( ) kπx and pug this into the wave equation to obtain ( ) kπx v (t)sin = c 2 v(t) k2 π 2 ( ) kπx 2 sin. This impies that if v = c2 k 2 π 2 v then u soves the wave equation. The compete soution of this ODE is 2 ( ) ( ) ckπ ckπ v(t) = α k cos t + β k sin t. v satisfies the Dirichet boundary conditions since sin() = sin(kπ) = for a k Z. The constants α k and β k are determined by the initia conditions, and we find that α k = a k and β k = ckπ b k, and hence [ ( ) ckπ u(x, t) = a k cos t + ( )] ( ) ckπ kπx ckπ b k sin t sin is a soution of the initia boundary-vaue probem (IVBP) (6), (7). By inearity, if we instead require the initia conditions then u(x, ) = φ(x) := u(x, t) = k=1 n ( ) kπx a k sin, u t (x, ) = ψ(x) := k=1 is a soution of the IBVP (6), (8). n ( kπx b k sin k=1 n [ ( ) ckπ a k cos t + ( )] ( ) ckπ kπx ckπ b k sin t sin ). (8)

10 8 MA25 Introduction to Partia Differentia Equations 4.2 Fourier Coefficients In the previous section we saw that when the initia conditions on the wave equation took a particuary nice form, we coud express the soution u as a sum of sines and cosines. However, we ony considered the case of finite sums: what happens when we take infinite sums? In this case we ca such series Fourier series. Given f C(R), there are two fundamenta questions: can we express it as a Fourier series, i.e. can we find a k, b k such that f(x) = k= [a k cos(kx)+b k sin(kx)], and when and how does this series converge to f(x)? We first answer the first question. Definition 4.2 (Fourier series). Let f : R R be 2π-periodic. A Fourier series of f is a series of the form ( f(x) = ak cos(kx) + b k sin(kx) ). k= The n th partia Fourier series is simpy the sum to n terms, i.e. f n (x) = n ( k= ak cos(kx)+b k sin(kx) ). It is often more convenient to consider compex-vaued functions f : R C, in which case the Fourier series takes the form f(x) = ˆf(k)e ikx. k= In this case the n th partia Fourier series is f n (x) = n k= n ˆf(k)e ikx, where the ˆf(k) are caed the Fourier coefficients. Note that f n (x+2π) = f n (x); this impies that f n (x) wi not converge to f(x) if f is not 2π-periodic. How do we find the so-caed Fourier coefficients ˆf(k)? It can be shown that ˆf(k) are given by the formua ˆf(k) = 1 2π e ikx f(x)dx. 4.3 L 2 Convergence of Fourier Series We turn to the question of when and how Fourier series converge. There are three principa ways in which Fourier series can converge: Definition 4.3. Let f n : [, 2π] R be a sequence of functions. We say (f n ) converges to f : [, 2π] R 1. pointwise if for each x [, 2π], im n f n(x) = f(x). 2. uniformy if im sup n x [,2π] f n (x) f(x) =. 3. in the mean-square sense (or in the L 2 sense) if im n f n (x) f(x) 2 dx =. Note that uniform convergence is the strongest form and impies both L 2 and pointwise convergence; in genera no other impication such as L 2 = pointwise hods. Generay pointwise convergence is the weakest and many theorems about convergence ony appy to uniform convergence. Consider first L 2 convergence 2. We first show that the formua ˆf(k) = 1 2π 2π e ikx f(x)dx for the Fourier coefficients is not arbitrary, but in fact minimises the L 2 distance between f and its partia Fourier series: 2 This is in some sense the most genera form of convergence; in fact the Fourier series of f converges to f in the L 2 sense provided ony that R 2π f(x) 2 dx is finite. The proof of this resut is a deep resut invoving the Lebesgue integra, and it essentiay stems from the space of a square-integrabe functions (i.e. functions f such that R 2π f(x) 2 dx < ) being compete; this is proved in MA359 Measure Theory.

11 MA25 Introduction to Partia Differentia Equations 9 Theorem 4.4. Let f C([, 2π], C). Among a possibe choices of 2n + 1 constants c n,...,c n the choice that minimises f(x) e ikx 2 c k dx is c k = ˆf(k) = 1 2π e ikx f(x)dx. k n One of the consequences of this is Besse s inequaity. Proposition 4.5 (Besse s inequaity). Let f C([, 2π], C), and et its Fourier coefficients be given by ˆf(k) = 1 2π 2π e ikx f(x)dx. Then 2π ˆf(k) 2 k Z f(x) 2 dx When we have equaity in Besse s inequaity, the Fourier series converges in the L 2 sense: Proposition 4.6 (Parseva s equaity). Let f C([, 2π], C), and et its Fourier coefficients be given by ˆf(k) = 1 2π 2π e ikx f(x)dx. The Fourier series of f converges in the L 2 sense, i.e. f(x) e ikx 2 2π ˆf(k) dx = im f(x) e ikx 2 ˆf(k) dx =, n k Z if and ony if 2π ˆf(k) 2 = k Z A consequence of Besse s inequaity is the foowing: f(x) 2 dx. k n Lemma 4.7 (Riemann Lebesgue Lemma). Let f C([, 2π], C). Then im k sin(kx)f(x)dx = im k cos(kx)f(x)dx = Proof. Assume, without oss of generaity, that f is rea-vaued. (Otherwise prove the Riemann Lebesgue Lemma for the rea and imaginary parts of f separatey.) sin(kx)f(x) dx 2πIm = ˆf(k) 2π ˆf(k) Since we know that k Z ˆf(k) 2 <, ceary im k ˆf(k) 2 =. 4.4 Pointwise and Uniform Convergence of Fourier Series We now consider the conditions required for uniform and pointwise convergence of Fourier series. If f is C 2, then the Fourier series converges uniformy: Theorem 4.8 (Uniform convergence of the Fourier series). Let f C 2 (R) be 2π-periodic. Then im f(x) e ikx ˆf(x) =, sup n x R i.e. the Fourier series of f converges uniformy to f. k n We can aso show that the Fourier series converges pointwise provided that the function is continuousy differentiabe (i.e. f C 1 ): Theorem 4.9 (Pointwise convergence of the Fourier series). Let f C 1 (R) be 2π-periodic. Then for each x [, 2π], im e ikx ˆf(k) = f(x), n k n i.e. the Fourier series of f converges pointwise to f.

12 1 MA25 Introduction to Partia Differentia Equations In fact, we can weaken the assumptions on f and sti (amost) get pointwise convergence: Theorem 4.1. Let f : R C be a 2π-periodic, piecewise-continuous function, i.e. there exists a finite set D [, 2π] such that f is continuous for a x [, 2π]\D). If at every point of discontinuity x D, the imits im x x f(x) and im x x + f(x) exist, then the partia Fourier series e ikx ˆf(k) converges pointwise to as n. k n f(x) for x R \ (D + 2πZ) 1 2 im f(x) + 1 x x 2 im f(x) for x (D + 2πZ) x x + That is, if f is piecewise continuous, then the Fourier series converges pointwise to the function, except at the jump discontinuities where it converges to the average of the imits from either side. At these jump discontinuities, the Fourier series overshoots by approximatey 9%; this is known as the Gibbs phenomenon. 5 The Lapace Operator 5.1 The Lapace Equations The fina PDE we study is aso the most important. First reca the definition of the Lapacian: Definition 5.1 (Lapacian). For a scaar fied u: R n R, the Lapacian of u, denoted u (or 2 u), is defined as n 2 u(x) := u(x). x 2 i=1 i In one dimension, u(x) = u xx (x); in two dimensions, u(x, y) = u xx (x, y) + u yy (x, y); in three dimensions, u(x, y, z) = u xx (x, y, z) + u yy (x, y, z) + u zz (x, y, z). For an open set Ω R n with boundary Ω, the Poisson equation is the PDE u(x) = f(x) where f C(Ω). Dirichet boundary conditions for the Poisson equation take the form u(x) = g(x) for a x Ω, where g C( Ω). The specia case u(x) = is caed the Lapace equation; the soutions of the Lapace equation are caed harmonic functions. Now we can adapt the maximum principe theorem for the Lapace equation. Theorem 5.2 (Maximum Principe). Let Ω R n be open and bounded by Ω and et u C 2 (Ω) be harmonic in Ω. Then u achieves its maximum at some point x Ω, i.e. max u(x) = max u( x). x Ω x Ω Proof. We can prove this much ike we proved the Maximum Principe for the diffusion equation. Fix ε > and et v(x) = u(x) + ε x 2 ; then v(x) = (u + ε x 2 ) = 2εn >. By the second derivative test, v(x) at an interior maximum point, hence v has no maximum in the interior of Ω. Since v is continuous, it must have a maximum in Ω, hence any maximum x must ie in Ω. Then for a x Ω, u(x) < v(x) v(x ) = u(x ) + ε x 2 max x Ω u( x) + εr2, where R is such that Ω {x R n x R}. Since ε was arbitrary, u(x) max x Ω u( x) for a x Ω, i.e. the maximum vaue of u is achieved on the boundary Ω.

13 MA25 Introduction to Partia Differentia Equations 11 The Lapace operator has the property that it doesn t change if we rotate the coordinate system. Because of this the form of the Lapacian is reativey simpe in both poar and spherica coordinates: Proposition 5.3 (Lapacian in poar coordinates). In poar coordinates in two dimensions, x = r cosθ and y = r sin θ, the Lapacian takes the form 2 u = u rr + 1 r u r + 1 r 2 u θθ = 2 u r u r r u r 2 θ 2. Proposition 5.4 (Lapacian in spherica coordinates). In spherica coordinates in three dimensions, x = r cosϕsin θ, y = r sinϕsin θ, z = r cosθ, the Lapacian takes the form 3 u = u rr + 2 r u r + 1 ( r 2 u θθ + (cotθ)u θ + 1 ) sin 2 θ u ϕϕ = 2 u r u r r + 1 ( 2 u r 2 θ 2 + (cotθ) u θ + 1 sin 2 θ We can use poar and spherica coordinates in a separation Ansatz: the technique of separation of variabes yieds very simpe resuts in the cases of circuar and spherica symmetry. 2 u ϕ 2 ). 5.2 The Submean Vaue Property of Harmonic Functions Definition 5.5 (Subharmonic). A function u: Ω R, where Ω R n is open, is sub-harmonic if u(x) for a x Ω. A function satisfying f ( tx + (1 t)y ) < tf(x) + (1 t)f(y) for a x, y R with x y and a t [, 1] is caed stricty convex; a stricty convex functions are sub-harmonic. Now we ook at the sub-mean vaue property of sub-harmonic functions. Definition 5.6. Let u: Ω R where Ω R 3 is open. The (surface) mean vaue u R (x ) of u on a ba B(x, R) Ω is defined by: u R (x ) := 1 4πR 2 u ds. B(x,R) Definition 5.7. u: Ω R has the sub-mean vaue property if whenever B(x, R) Ω then u(x ) u R (x ) = 1 4πR 2 u ds B(x,R) i.e. the vaue of u at the centre of a ba is ess than or equa to the average vaue of u of the boundary of the ba. Theorem 5.8. Let Ω R n be open. If u C 2 (Ω) has u (i.e. u is sub-harmonic) and B(x, R) Ω, then u has the sub-mean vaue property, i.e. u(x ) u R (x ). Note that if u is harmonic then u and u are both sub-harmonic, and hence u(x ) = u R (x ) whenever B(x, R) Ω. This aows us to prove the Strong Maximum Principe, which says not ony that the maximum occurs on the boundary Ω, but that it cannot occur inside Ω uness u is constant. Proposition 5.9 (Strong Maximum Principe). Let Ω R n be open, connected and bounded, and et u C 1 (Ω) be harmonic. Then u achieves its maximum in Ω if and ony if u is constant.

14 12 MA25 Introduction to Partia Differentia Equations The ast property of harmonic functions we study is Dirichet s principe: Theorem 5.1 (Dirichet s Principe). Let Ω R n be open and bounded and et u C 1 (D) be harmonic subject to the boundary condition u(x) = f(x) for a x Ω. Then for any v C 1 (D) satisfying the same boundary condition, i.e. v(x) = f(x) for a x Ω, we have 1 2 Ω u(x) 2 dx 1 2 Ω v(x) 2 dx. That is, of a functions u C 1 (D) subject to the boundary condition u(x) = f(x) for a x Ω, the one which minimises the energy integra E(u) := 1 2 Ω u(x) 2 dx is harmonic. 5.3 Types of Second-Order PDEs We have studied three very different types of second-order inear PDEs, namey the wave equation, diffusion equaition and the Lapace equation. It turns out that we can reduce every other second-order inear PDE to one of these three equations, moduo ower order terms (.o.t.). Theorem 5.11 (Types of second order differentia equation). Consider the genera PDE a 11 u xx + 2a 12 u xy + a 22 u yy + a 1 u x + a 2 u y + a u =. 1. If a 2 12 < a 11 a 22 the PDE is caed eiptic and is reducibe to u xx + u yy +.o.t. =. 2. If a 2 12 > a 11 a 22 the PDE caed hyperboic and can be reduced to u xx u yy +.o.t. =. 3. If a 2 12 = a 11 a 22 then the PDE caed paraboic and can be reduced to u xx +.o.t. =. The prototypica eiptic equation is the Lapace equation; the prototypica hyperboic equation is the wave equation; and the prototypica paraboic equation is the diffusion equation. Cosing Remarks As you can see, there s not too materia to PDEs, but there is scope for getting a bit confused, particuary between the different types of equation. Water Strauss s Partia Differentia Equations: An Introduction has iteray hundreds of questions for those wanting practice, as we as fiing any gaps in your theoretica knowedge. Above a practising ots of questions is the ony way to do we, so practise, practise, practise, and good uck in the exam!