Partial Differential Equations: An Introduction, 2nd Edition CONTENTS (The starred sections form the basic part of the book.) Chapter 1/Where PDEs Come From 1.1* What is a Partial Differential...

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Partial Differential Equations: An Introduction, 2nd Edition CONTENTS (The starred sections form the basic part of the book.) Chapter 1/Where PDEs Come From 1.1* What is a Partial Differential Equation? 1 1.2* First-Order Linear Equations 6 1.3* Flows, Vibrations, and Diffusions 10 1.4* Initial and Boundary Conditions 20 1.5 Well-Posed Problems 25 1.6 Types of Second-Order Equations 28 Chapter 2/Waves and Diffusions 2.1* The Wave Equation 33 2.2* Causality and Energy 39 2.3* The Diffusion Equation 42 2.4* Diffusion on the Whole Line 46 2.5* Comparison of Waves and Diffusions 54 Chapter 3/Reflections and Sources 3.1 Diffusion on the Half-Line 57 3.2 Reflections of Waves 61 3.3 Diffusion with a Source 67 3.4 Waves with a Source 71 3.5 Diffusion Revisited 80 Chapter 4/Boundary Problems 4.1* Separation of Variables, The Dirichlet Condition 84 4.2* The Neumann Condition 89 4.3* The Robin Condition 92 viii CONTENTS ix Chapter 5/Fourier Series 5.1* The Coefficients 104 5.2* Even, Odd, Periodic, and Complex Functions 113 5.3* Orthogonality and General Fourier Series 118 5.4* Completeness 124 5.5 Completeness and the Gibbs Phenomenon 136 5.6 Inhomogeneous Boundary Conditions 147 Chapter 6/Harmonic Functions 6.1* Laplace’s Equation 152 6.2* Rectangles and Cubes 161 6.3* Poisson’s Formula 165 6.4 Circles, Wedges, and Annuli 172 (The next four chapters may be studied in any order.) Chapter 7/Green’s Identities and Green’s Functions 7.1 Green’s First Identity 178 7.2 Green’s Second Identity 185 7.3 Green’s Functions 188 7.4 Half-Space and Sphere 191 Chapter 8/Computation of Solutions 8.1 Opportunities and Dangers 199 8.2 Approximations of Diffusions 203 8.3 Approximations of Waves 211 8.4 Approximations of Laplace’s Equation 218 8.5 Finite Element Method 222 Chapter 9/Waves in Space 9.1 Energy and Causality 228 9.2 The Wave Equation in Space-Time 234 9.3 Rays, Singularities, and Sources 242 9.4 The Diffusion and Schrödinger Equations 248 9.5 The Hydrogen Atom 254 Chapter 10/Boundaries in the Plane and in Space 10.1 Fourier’s Method, Revisited 258 10.2 Vibrations of a Drumhead 264 10.3 Solid Vibrations in a Ball 270 10.4 Nodes 278 10.5 Bessel Functions 282 x CONTENTS 10.6 Legendre Functions 289 10.7 Angular Momentum in Quantum Mechanics 294 Chapter 11/General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 299 11.2 Computation of Eigenvalues 304 11.3 Completeness 310 11.4 Symmetric Differential Operators 314 11.5 Completeness and Separation of Variables 318 11.6 Asymptotics of the Eigenvalues 322 Chapter 12/Distributions and Transforms 12.1 Distributions 331 12.2 Green’s Functions, Revisited 338 12.3 Fourier Transforms 343 12.4 Source Functions 349 12.5 Laplace Transform Techniques 353 Chapter 13/PDE Problems from Physics 13.1 Electromagnetism 358 13.2 Fluids and Acoustics 361 13.3 Scattering 366 13.4 Continuous Spectrum 370 13.5 Equations of Elementary Particles 373 Chapter 14/Nonlinear PDEs 14.1 Shock Waves 380 14.2 Solitons 390 14.3 Calculus of Variations 397 14.4 Bifurcation Theory 401 14.5 Water Waves 406 Appendix A.1 Continuous and Differentiable Functions 414 A.2 Infinite Series of Functions 418 A.3 Differentiation and Integration 420 A.4 Differential Equations 423 A.5 The Gamma Function 425 References 427 Answers and Hints to Selected Exercises 431 Index 446 1 WHERE PDEs COME FROM After thinking about the meaning of a partial differential equation, we will flex our mathematical muscles by solving a few of them. Then we will see how naturally they arise in the physical sciences. The physics will motivate the formulation of boundary conditions and initial conditions. 1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? The key defining property of a partial differential equation (PDE) is that there is more than one independent variable x, y, . . . . There is a dependent variable that is an unknown function of these variables u(x, y, . . . ). We will often denote its derivatives by subscripts; thus ∂u/∂x = ux , and so on. A PDE is an identity that relates the independent variables, the dependent variable u, and the partial derivatives of u. It can be written as F(x, y, u(x, y), ux (x, y), uy(x, y)) = F(x, y, u, ux , uy) = 0. (1) This is the most general PDE in two independent variables of first order. The order of an equation is the highest derivative that appears. The most general second-order PDE in two independent variables is F(x, y, u, ux , uy, uxx , uxy, uyy) = 0. (2) A solution of a PDE is a function u(x, y, . . . ) that satisfies the equation identically, at least in some region of the x, y, . . . variables. When solving an ordinary differential equation (ODE), one sometimes reverses the roles of the independent and the dependent variables—for in- stance, for the separable ODE du dx = u3. For PDEs, the distinction between the independent variables and the dependent variable (the unknown) is always maintained. 1 2 CHAPTER 1 WHERE PDEs COME FROM Some examples of PDEs (all of which occur in physical theory) are: 1. ux + uy = 0 (transport) 2. ux + yuy = 0 (transport) 3. ux + uuy = 0 (shock wave) 4. uxx + uyy = 0 (Laplace’s equation) 5. utt − uxx + u3 = 0 (wave with interaction) 6. ut + uux + uxxx = 0 (dispersive wave) 7. utt + uxxxx = 0 (vibrating bar) 8. ut − iuxx = 0 (i = √−1) (quantum mechanics) Each of these has two independent variables, written either as x and y or as x and t. Examples 1 to 3 have order one; 4, 5, and 8 have order two; 6 has order three; and 7 has order four. Examples 3, 5, and 6 are distinguished from the others in that they are not “linear.” We shall now explain this concept. Linearity means the following. Write the equation in the form lu = 0, wherel is an operator. That is, if v is any function,lv is a new function. For instance, l = ∂/∂x is the operator that takes v into its partial derivative vx . In Example 2, the operator l is l = ∂/∂x + y∂/∂y. (lu = ux + yuy.) The definition we want for linearity is l(u + v) = lu + lv l(cu) = clu (3) for any functions u, v and any constant c. Whenever (3) holds (for all choices of u, v, and c), l is called linear operator. The equation lu = 0 (4) is called linear if l is a linear operator. Equation (4) is called a homogeneous linear equation. The equation lu = g, (5) where g �= 0 is a given function of the independent variables, is called an inhomogeneous linear equation. For instance, the equation (cos xy2)ux − y2uy = tan(x2 + y2) (6) is an inhomogeneous linear equation. As you can easily verify, five of the eight equations above are linear as well as homogeneous. Example 5, on the other hand, is not linear because although (u + v)xx = uxx + vxx and (u + v)t t = utt + vt t satisfy property (3), the cubic term does not: (u + v)3 = u3 + 3u2v + 3uv2 + v3 �= u3 + v3. 1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? 3 The advantage of linearity for the equation lu = 0 is that if u and v are both solutions, so is (u + v). If u1, . . . , un are all solutions, so is any linear combination c1u1(x) + · · · + cnun(x) = n∑ j=1 c j uj (x) (cj = constants). (This is sometimes called the superposition principle.) Another consequence of linearity is that if you add a homogeneous solution [a solution of (4)] to an inhomogeneous solution [a solution of (5)], you get an inhomogeneous solu- tion. (Why?) The mathematical structure that deals with linear combinations and linear operators is the vector space. Exercises 5–10 are review problems on vector spaces. We’ll study, almost exclusively, linear systems with constant coefficients. Recall that for ODEs you get linear combinations. The coefficients are the arbitrary constants. For an ODE of order m, you get m arbitrary constants. Let’s look at some PDEs. Example 1. Find all u(x, y) satisfying the equation uxx = 0. Well, we can integrate once to get ux = constant. But that’s not really right since there’s another variable y. What we really get is ux(x, y) = f (y), where f (y) is arbitrary. Do it again to get u(x, y) = f (y)x + g(y). This is the solution formula. Note that there are two arbitrary functions in the solution. We see this as well in the next two examples. � Example 2. Solve the PDE uxx + u = 0. Again, it’s really an ODE with an extra variable y. We know how to solve the ODE, so the solution is u = f (y) cos x + g(y) sin x, where again f (y) and g(y) are two arbitrary functions of y. You can easily check this formula by differentiating twice to verify that uxx = −u. � Example 3. Solve the PDE uxy = 0. This isn’t too hard either. First let’s integrate in x, regarding y as fixed. So we get uy(x, y) = f (y). Next let’s integrate in y regarding x as fixed. We get the solution u(x, y) = F(y) + G(x), where F ′ = f. � 4 CHAPTER 1 WHERE PDEs COME FROM Moral A PDE has arbitrary functions in its solution. In these examples the arbitrary functions are functions of one variable that combine to produce a function u(x, y) of two variables which is only partly arbitrary. A function of two variables contains immensely more information than a function of only one variable. Geometrically, it is obvious that a surface {u = f (x, y)}, the graph of a function of two variables, is a much more com- plicated object than a curve {u = f (x)}, the graph of a function of one variable. To illustrate this, we can ask how a computer would record a function u = f (x). Suppose that we choose 100 points to describe it using equally spaced values of x : x1, x2, x3, . . . , x100. We could write them down in a column, and next to each xj we could write the corresponding value uj = f (xj ). Now how about a
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Answer To: Partial Differential Equations: An Introduction, 2nd Edition CONTENTS (The starred sections form the...

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