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MA319 Partial Differential Equations Amol Sasane

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MA319 Partial Differential Equations Amol Sasane MA319 Partial Differential Equations Amol Sasane Contents Introduction 1 What is a Partial Differential Equation? 1 Why study PDEs? 9 What does “study” mean? 9 Function spaces, operators, solutions 10 Order of a PDE 11 Linear versus nonlinear 11 Homogeneous versus nonhomogeneous 12 The classical trinity 13 Initial Boundary Value Problems 15 Some historical remarks 17 Chapter 1. First Order PDEs 19 §1.1. An example of deriving a PDE: traffic flow 19 §1.2. The method of characteristics 22 Chapter 2. The classical trinity 29 §2.1. Classification of second order linear PDEs 29 §2.2. Uniqueness and stability 31 §2.3. Discretization and the finite difference method 45 Chapter 3. Separation of variables 49 §3.1. Fourier series 49 §3.2. Dimensionless form 54 §3.3. Diffusion equation with Dirichlet conditions 55 §3.4. Diffusion equation with Neumann conditions 62 v vi Contents §3.5. Wave equation with Dirichlet conditions 65 §3.6. Laplace equation 74 §3.7. Inhomogeneous Dirichlet and Neumann problems 77 p˚q §3.8. Proof of the Fourier Series Theorem 83 Chapter 4. Integral transform methods 89 §4.1. Fourier transform 90 §4.2. The Diffusion equation 96 §4.3. Application: European option pricing 101 §4.4. Dirichlet’s problem for a half plane 105 §4.5. Oscillations of an infinite string 109 §4.6. The Laplace transform method 112 §4.7. Proof of the Fourier Transform Theorem 119 Chapter 5. Distributions and weak solutions 121 §5.1. Test functions, distributions, and examples 122 §5.2. Derivatives in the distributional sense 126 §5.3. Weak solutions 128 §5.4. Fourier transform 133 Solutions 135 Solutions to the exercises from the Introduction 135 Solutions to the exercises from Chapter 1 148 Solutions to the exercises from Chapter 2 152 Solutions to the exercises from Chapter 3 162 Solutions to the exercises from Chapter 4 175 Solutions to the exercises from Chapter 5 186 Bibliography 193 Index 195 Introduction What is a Partial Differential Equation? A Partial Differential Equation (PDE) is an equation involving known and unknown functions of several real variables and their partial derivatives. So a PDE is analogous to an ODE (Ordinary differential equation, which is an equation involving known and unknown functions of one variable t, usually thought of as time), except that now we have functions of several variables (say, t,x1, ,xn). ¨ ¨ ¨ Exercise 0.1 (ODE background). In our study of PDEs, we will need the knowledge of solutions to the following two ODEs, and the aim of this exercise is to quickly recall this. (1) (First order linear ODE). Let a : R R be a continuous function. Show that the initial value problem Ñ x1 t a t x t , t R, p q“ p q p q P x ti xi R. " p q“ P t has a unique solution, given by x t exp a τ dτ xi,t R. p q“ t p q ¨ P ´ ż i ¯ (2) (Second order linear ODE). Let λ R. The aim of this exercise is to show that the equation P x2 t λx t 0, x 0 A and x1 0 B (0.1) p q` p q“ p q“ p q“ has the unique solution given by B A cos kt sin kt if λ k2, k 0 p q` k p q “ ą $ x t ’ A Bt if λ 0 p q“ ’ ` “ &’ B 2 A cosh kt sinh kt if λ k , k 0. k ’ p q` p q “´ ą ’ To see this, first%’ verify that the x given by the formula above is a solution. One can show that if A, B are zero, then 0 is the only solution, as follows. First note that by virtue of the equation x2 λx 0, it is clear that x is infinitely many 2n ` n“n 2n 1 n 1 n times differentiable. Moreover, xp q 1 λ x and xp ´ q 1 ´ λ x. Since “ p´ q “ p´ q 1 2 Introduction k x 0 x1 0 0, it follows that xp q 0 0 for all k. Using a Taylor polynomial of degreep q “2np q1 “, Taylor’s Formula givesp theq “ existence of a θ 0, 1 such that ´ P p q 2n xp q θt 2 x t p qt n. p q“ 2n ! p q Now suppose we choose any c 0, and let M 0 be such that x t M on c,c . 2n n ą ą | p q| ď r´ s Then xp q t λ M on c,c . Thus | p q| ď | | r´ s n 2n λ M 2n λ c n 0 x t | | c M p | | q Ñ8 M 0 0. ď | p q| ď 2n ! “ ¨ 2n ! ÝÑ ¨ “ p q ap q Thus x 0 on c,c . But as c 0 was arbitrary, we conclude that x 0 on R. The” uniquenessr´ tos (0.1) nową follows immediately by just considering” the difference of two possible solutions. Here are a few examples of PDEs. Example 0.1 (Transport Equation). Consider u u B x,t B x,t 0. t p q` xp q“ u B B The function x,t u x,t is the unknown function. In the sequel, it will be conve- p q ÞÑ p q nient to use the notation u ut : B , “ t Bu ux : B . “ x B The transport equation can then be written simply as ut ux 0. ♦ ` “ In general, if w is a smooth enough function of the variables x1,x2, ,xn (and possi- ¨ ¨ ¨ bly several others), then it will be convenient to use the following notation: nw wx1 xn : B . ¨¨¨ “ x1 xn B ¨¨¨B Example 0.2 (Mathematical physics). Most of the equations in Mathematical Physics are PDEs. So PDEs describe our physical universe! For example in (nonrelativistic) quantum mechanics, the fundamental equation is the Schrödinger equation. For the hydrogen atom, this equation is ℏ2 e2 iℏΨt Ψxx Ψyy Ψzz Ψ, ´ “ 2mp ` ` q` x2 y2 z2 ` ` where a m is the mass of the electron, e is the charge of the electron, ℏ is h 2π, and h is Planck’s constant, { Ψ is the “wave function”. What is a Partial Differential Equation? 3 Note that this is a PDE. The quantity Ψ 2dxdydz | | ¡Ω gives the probability of finding the electron in the region Ω R3 at time t. (This equa- Ă tion can be used for example to explain the energy levels of the electron in the hydrogen atom. Variations of the equation explain the structure of all atoms and molecules, and so all of Chemistry!) As opposed to the above PDE example, with just one equation, one can also have a system of PDEs where there are m 1 equations in ℓ 1 unknowns. A classical ą ą example is the Maxwell equations1 of electromagnetism describing the evolution of the electric field E and the magnetic field B. ∇ E ρ ∇ ¨ “ B ...and there was light! E B t ∇ ˆB “´0 B ∇ ¨ “ E B j B t ˆ “ ` B In Einstein’s theory of General Relativity, describing spacetime, the “spacetime metric” is obtained as a solution to a PDE called the Einstein field equation. ♦ Exercise 0.2 (Schrödinger equation). (1) (Probability is conserved). Consider the free one dimensional Schrödinger equation ℏ2 iℏΨt Ψxx. “´2m Show that for any C8 solution Ψ: R 0, C such that Ψ ,t has compact support for all t, ˆr 8q Ñ p¨ q 8 8 Ψ x, t 2dx Ψ x, 0 2dx. | p q| “ | p q| ż´8 ż´8 (2) The one dimensional Schrödinger equation is ℏ2 iℏΨt Ψxx V x Ψ, “´2m ` p q ¨ 1See Exercise 0.3 for an explanation of the notation ∇ and ∇ . ˆ ¨ 4 Introduction where V is the potential energy. One method to find solutions is to assume that variables separate, that is, the solution has the form Ψ x, t X x T t . p q“ p q p q Substituting this in the Schrödinger equation gives ℏ2 iℏT 1 t 2 X2 x V x X x p q ´ m p q` p q p q . T t “ X x p q p q As the left hand side depends only on t and the right hand side only on x, the only way equality can occur is if both sides are equal to the same constant, say E (for “energy”). So we obtain the equation following for T , iE T 1 t ´ T t , p q“ ℏ p q which has the solution iEt Et Et T t C exp ´ C cos i sin . p q“ ℏ “ ℏ ´ ℏ ´ ¯ ˆ ´ ¯ ´ ¯˙ The equation for X is ℏ2 X2 x V x E X x 0. ´2m p q ` p p q´ q p q“ Consider a free particle of mass m confined to the interval 0 x π (so that V 0 in 0, π ), and suppose that Ψ 0,t Ψ π,t 0 for all t. (Imagineă ă the particle to” be inp an “infiniteq potential well”.)p Showq“ thatp thisq“ problem has a nontrivial solution if and only if n2ℏ2 E , n 1, 2, 3, . “ 2m “ ¨ ¨ ¨ Sketch the probability density function Ψ 2 when n 1, 2, and compute the probabil- ity that the particle is in the interval 0,|1 |4 in each“ case. r { s Exercise 0.3 (curl:=∇ and div:=∇ ). The aim of this exercise is to introduce the fundamental operators called curl andˆ divergence¨ from vector calculus. A vector field v : R3 R3 is just a map which assigns to every point x R3 a vector v x .
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