Partial Differential Equations - part of EM Waves module (PHY2065)

Transcription

1 Partial Differential Equations - part of EM Waves module (PHY2065) Richard Sear February 7, 2013 Recommended textbooks 1. Mathematical methods in the physical sciences, Mary Boas. 2. Essential mathematical methods for the physical sciences, Riley and Hobson. 3. Advanced Engineering Mathematics, Erwin Kreyszig All 3 are in the library. Riley and Hobson is also available as an ebook. Overview 1. Introduction to PDEs: more than one variable; the wave equation; Laplace s equation etc; boundary conditions (BCs). 2. The wave equation (in one dimension). 3. The diffusion equation (two dimensions, steady state). 4. Spherical and circular polar coordinates. 5. Schrödinger equation for the hydrogen atom (s wavefunctions spherically symmetric solutions). 1

2 1 Introduction to Partial Differential Equations Partial differential equations (PDEs) differ from ordinary differential equations (ODEs) in that there is more than one variable. Thus the derivatives are partial derivatives, and so they are called partial differential equations. In physics when we have more than one variable this is almost always either because the functions depend on more than one spatial dimension, e.g., not on x only, but on say both x and y, or because they depend on space and time, e.g., x and t. As they contain more than one variable, PDEs can be trickier than ODEs, and in both cases unless they are linear they are often impossible to solve analytically nonlinear equations are typically solved using a computer. Here we will only consider linear PDEs. In this part of the course, we will introduce the most common simple PDEs in physics, and show how to solve them. As it happens these are all second-order, i.e., contain second derivatives, and so we will only study second-order PDEs. 1.1 Time-independent PDEs Firstly, there are three common named PDEs that depend on the position coordinates x, y and z, but not on time. For a scalar function u(x,y,z) these are: Laplace s equation 2 u = 0 Poisson s equation 2 u = f(x,y,z) with f some specified function of x and y and z, e.g., f = x 2 + 2y 2 + z 2, and Helmholtz s equation 2 u + au(x,y,z) = 0 with a a constant. They all include 2 which is in three dimensions 2 = 2 x y z 2 Thus in three dimensions Laplace s equation is ( ) 2 2 u = x y + 2 u = 2 u 2 z 2 x + 2 u 2 y + 2 u 2 z = 0 2 In, for example, one dimension, this simplifies to d 2 u(x) dx 2 = 0 where the derivatives are no longer partial derivatives as u(x) is now a function only of x. Note that Poisson s equation is the most general, as if f = 0 it reduces to Laplace s equation and if f = au it reduces to Helmholtz s equation. Poisson s equation appears in electromagnetism. There u = φ(x, y, z) the electrostatic potential, and f = ρ(x,y,z)/ǫ 0, with ρ the charge density as a function of position and ǫ 0 the permittivity. Then Poisson s equation is 2 φ = ρ/ǫ 0 2

3 which is one of Maxwell s four equations that govern electromagnetism, in the electrostatic case. Thus, Poisson s equation occurs very frequently in electromagnetism. Also, in a region of space that is free of charges, i.e., where ρ = 0, then Poisson s equation for the electrostatic potential φ simplifies to 2 φ = 0 which is Laplace s equation. Thus Laplace s equation also appears frequently in electromagnetism. Systems like semiconductors, salty water etc have free charges, electrons and/or ions. At thermal equilibrium the charge density of these charges ρ is often is proportional to the electrostatic potential. Then we have ρ φ and Poisson s equation becomes Helmholtz s equation. Thus Helmholtz s equation is common whenever we are dealing with electrons or ions. Another important time-independent PDE that you have already come across is Schrödinger s time-independent equation for the quantum mechanical wavefunction ψ 1.2 Time-dependent PDEs 2 2m 2 ψ(x,y,z) + V (x,y,z)ψ(x,y,z) = Eψ(x,y,z) So much for the time independent PDEs. Time dependent PDEs need to be solved for the function of position and time v(x,y,z,t). Most of the common simple time dependent PDEs in physics are either of the form of the diffusion equation or the wave equation. The diffusion equation is diffusion equation D 2 v = v t where D is a constant, the diffusion constant. D has dimensions of a length squared over time. The wave equation is wave equation 2 v = 1 2 v c 2 t 2 c is the speed of the wave and has dimensions of length over time. Note that the difference between the two PDEs is that the diffusion equation has the first derivative with respect to time, while the wave equation has the second derivative with respect to time. As we will see, this apparently small difference will result in their solutions being very different. As their names suggest the diffusion equation describes diffusion, e.g., of molecules in a gas or liquid, whereas the wave equation describes waves: light, sound etc. Also note that Schrödinger s time-dependent equation looks very like a diffusion equation. For the time-dependent wave function ψ(x,y,z,t) of an electron in a potential V (x,y,z) it is 2 2m 2 ψ(x,y,z,y) + V (x,y,z)ψ(x,y,z,t) = i ψ(x,y,z,t) t If the potential V = 0 then the equation is the diffusion equation i 2m 2 ψ(x,y,z,t) = ψ(x,y,z,t) t where we divided by i. This is almost the diffusion equation, but with D = i /2m, i.e., with an imaginary diffusion constant. This looks a bit strange but the maths works out fine. 3

4 Also, when we study diffusion, sometimes, we are studying it at steady-state, which means (by definition) that v is not a function of time. This is typically diffusion down a constant concentration gradient. Then as v is not a function of time, the time derivative is of course zero and the diffusion equation reduces to Laplace s equation. Thus, Laplace s equation also occurs whenever diffusion occurs down a time-independent concentration gradient. 1.3 Boundary conditions for PDEs Time-independent PDEs To place the BCs for second-order PDEs in context, let us start by recalling how the BCs work for second-order ODEs. For a second order ODE we require two boundary conditions, typically either the value of the function at two points or the value of the function and its first derivative at a point. For example, if we have the general solution of an ODE, call it f(x), and want to know f(x) for x between 0 and 1, then a possible set of boundary conditions consists of the values of f(x) at both x = 0 and x = 1 i.e., the values of f at the boundaries of the range of x values we are interested in. The boundary conditions for a PDE are analogous except that if we have a function of, say, two variables x and y, f(x,y), then we will be interested not in a function along a line but a function over an area of the xy plane. This is illustrated in Fig. 1, where we show a possible area A over which we want the solution f(x,y) to a PDE. See the caption for possible BCs for that area. A particularly simple area to work with is that of a rectangle. A rectangular area is bounded by four straight lines. Once we have the general solution of a PDE, f(x,y), then possible boundary conditions are the values of the function f(x,y) along all four sides of the rectangle. Other possible BCs are the values of the derivative of f(x,y) along these four sides. This derivative of f(x,y) must be the derivative normal to the direction of the boundary 1, at all points along this boundary. It is also possible to have boundary conditions that combine values of f and its derivative along the boundary of the area. Here either f or its derivative must be specified at every point on the boundary. An example set of BCs for the rectangle is that f(x,y) = 1 along all four sides of the rectangle. In three dimensions, we have three variables, x, y and z, and then we will require the function f(x,y,z) over some volume and the boundary conditions are be the values of the function or its derivative over the surface that encloses this volume. For example, if we want the f(x,y,z) inside a cubic volume then one possible BC is the function s value over all six faces of this cube. One final small point, if along the boundary of an area (or a volume in three dimensions) we only specify the derivative (i.e., we don t specify the function itself anywhere), then the function is only specified up to an unknown constant, i.e., we end up with f(x,y) + C with f precisely defined and C an unknown constant. Often not knowing C does not matter Time-dependent PDEs For time-dependent PDEs, the boundary conditions often include initial conditions, i.e., the function at time t = 0. For example, for the wave equation, the boundary condition might be the function f(x,t) and its time derivative, at t = 0. If the function is defined all along the x axis that may be enough, but sometimes boundary conditions at the ends of some length of the x axis are imposed as well. For example, if the wave is on a taught string, the boundary conditions may be fixed values of 1 Recall from your vector calculus lectures that the derivative of a scalar f in two or three dimensions, f, is a vector. 4

5 Figure 1: Schematic of the xy plane, to illustrate boundary conditions for a time-independent PDE. In two dimensions we will be interested in the solution, f(x,y), within some area A. A possible area A is shown as the yellow (grey) shaded area bounded by the black curve. The boundary conditions for a solution of f(x,y) in A can then be: 1) the values of function f(x,y) along the complete boundary (shown as black curve); 2) the values of the derivative of f(x,y) normal to the boundary, along the complete boundary; or 3) some combination of f and its derivative, along the complete boundary. y area A f(x,y) x f at the two ends of the string, applied at at all times, plus the position and velocity of the string at time t = 0. In general, the BCs here can be quite varied, but in this course only a few simple examples will be discussed. I do not expect you to know the general definition of the BCs of a time-dependent PDE Superposition for homogeneous PDEs When you studied ODEs, you came across homogeneous ODEs. These were ODEs in which all terms were linear in either the function itself or a derivative of the function, i.e., if the function is F(x), each term is proportional to F(x) or df(x)/dx, etc, and there are no F 2, F 3, exp(f), (df(x)/dx) 2, etc terms, and no terms that are just functions of x. Homogeneous ODEs have the very useful property that the sum of any two solutions of the ODE, is also a solution to the ODE, i.e., if F 1 (x) and F 2 (x) are solutions are an ODE, then a 1 F 1 (x)+a 2 F 2 (x) is also a solution to the same ODE. Here a 1 and a 2 are any two constants. Homogeneous PDEs are exactly analogous to homogeneous ODEs. They are PDEs in which every term is proportional to F(x,y,z,t) or df(x)/dx, d 2 F(x)/dxdy, etc, and there are no F 2, F 3, exp(f), (df(x)/dx) 2, etc terms, and no terms that are just functions of x, y, z and/or t. Homogeneous PDEs have the same useful property as homogeneous ODEs. The sum of any two solutions of a homogeneous PDE is also a solution to the same PDE, i.e., if F 1 (x,y,z,t) and F 2 (x,y,z,t) are both solutions to a PDE, then F(x,y,z,t) = a 1 F 1 (x,y,z,t) + a 2 F 2 (x,y,z,t) F, F 1 and F 2 solutions of a homogeneous PDE is also a solution to the same PDE. This is true for any values of the constants a 1 and a 2. This property is extremely useful. In particular, often sines and cosines are solutions of a homogeneous PDE, and then we can construct a solution of the PDE that satisfies the imposed BCs, by summing lots of sine and cosine wave solutions to the PDE (which works because the PDE is homoge- 5

6 neous). This is what is called a Fourier series solution of a PDE, and these solutions are very widely used and very useful in the study of physics problems with PDEs. 6