18.15 - Introuction to PEs, Fall 004 Prof. Gigliola Staffilani Lecture - First orer linear PEs an PEs from physics I mentione in the first class some basic PEs of first an secon orer. Toay we illustrate how they come naturally as a moel of some basic phenomena. 1. u t + cu x = 0 Transport equation (simple transport). u tt c u xx = 0 Wave equation (vibrating string) 3. u t = kδu Parabolic equation (heat equation an iffusion) 4. Δu = 0 Elliptic equation (stationary wave an iffusion) 5. iu t = Δu Schröinger equation (Hyrogen atom) erivation of (1): Imagine putting a rop of ink in the pipe. Let u(x, t) be the concentration or ensity of ink at point x at time t. But how o we escribe it? Fix an interval [0, b]. b ink in [0, b] = M = u(x, t)x time=t 0 Suppose the water moves at spee c, so at time h + t the same quantity of ink will be b M = ch+b ch ch+b u(x, t + h)x u(x, t)x = 0 ch Taking b u(b, t) = u(b + ch, t + h) u(x, t + h)x Taking h h=0 0 = cu x (b, t) + u t (b, t) 1
erivation of (): A typical example of wave motion in a plane is the motion of a string with fixe en points. Why oes it have this shape? We will see later. Let s take a piece of it: We will ignore all the forces on the string except for its tension. We will consier a perfect string in which for all x we have T(x, t) = T v(x, t) (1) for constant T an v(x, t) a unit tangent vector to the string at (x, t). v(x, t) = (1, u x ) 1 + ux We assume two facts: 1. The string has constant mass ensity ρ.. The tension oes not change on the longituinal (x) component. Now fix an interval of the string [x 0, x 1 ] an use Newton s law: T(x 1, t) T(x 0, t) = ma, T > 0 1 x 1 T = 0 () 1 + u x x 0 x u 1 x x 1 T = ρu tt (x, t)x 1 + ux x 0 x0
Approximation: u x 1. Now consier the function f(z) = (1 + z ) 1/ an z 0. We can expan f(z) near z = 0: So f(z) = (1 + z ) 1/ 1 f (z) = (1 + z ) 3/ z, f (0) = 0 3 f (z) = z(1 + z ) 5/ (1 + z ) 3/, f (0) = 1 f(z) = f(0) + f f (0) (0)z + z +... 1 f(z) = 1 z +... so for z small we can replace f(z) by 1 an then () becomes T (x 1, t) T (x 0, t) = 0 (T constant with respect to x) x 1 x 1 T u x = ρu tt (x, t)x (3) x0 x0 Let s assume also that T = C is also constant with respect to time (t) an ifferentiate (3) with respect to x: (T u x ) x = ρu tt (x, t) u xx = c u tt, c = erivation of heat flow (3): Let R 3 be a region in space an let U(t) be the amount of energy in region. (Heat refers to a change in energy of a region.) Let u(x, y, z, t) be the temperature. Then we can write: U(t) = cρu(x, y, z, t)v ρ T where ρ is the ensity of (mass/volume) an c is the specific heat, (V x y z). 3
Clearly U(t) = t cρu t V. (Here we are assuming that the material of oes not change in time.) Assume, that outsie the boy the space is coler, an thus that the energy flux (irection of heat flow) will go from insie to outsie an will iffuse through the bounary of = So also we have U(t) = κ(n u)s, t which is the heat transfere through the surface, an κ > 0 is the thermal conuctivity. By the ivergence Theorem κ(n u)s = (κ u)v so If κ is constant, it follows that κ u t = Δu, cρ cρu t = (κ u) Δ = x + y + z. erivation of stationary wave or iffusion equation (4): Assume that in the heat an wave equations there is no change in time. The wave an heat equations reuce to Δu = 0 efinition: Any function u such that Δu = 0 is calle harmonic. Remark: In one imension all harmonic functions are linear: f xx = 0 f x = c 0 f = c 0 x + c 1 Schröinger Equation for the Hyrogen Atom (5): e i u t = Δu + u m r where is Planck s constant over π, an e /r is the potential energy of the system. Quantum mechanics tells us that u is a possible state (wavefunction) of the electron. 4
Let be a region in R 3. Then u (x, y, z, t)v is the probability that the electron is in at time t, R 3 z u (x, y, z, t)v is the expectation value of the electron z coorinate at time t, if we normalize u such that (We can perform similar calculations for x, y.) R 3 u (x, y, z, t)v = 1. (4) The expecte z-coorinate of the momentum is u V i (x, y, z, t) ū(x, y, z, t) z Remark: It is possible to normalize u in the way escribe above, an furthermore the normalization of u oes not change over time. Proof: uuv = Re u t ūv t R 3 R 3 i i e u t = i 1 ( i u t ) = Δu + u m r Re u t uv = Im R 3 R 3 Δu uv + Im 1 e u uv. m R 3 r }{{}}{{} ( ) =0 Since (Δu)ū = u u, an the latter is real, ( ) is zero as well. Exercise: Carefully erive the equation of motion for a string in a meium in which the 5
resistance is proportional to the velocity: Initial an Bounary Conitions Velocity v(x, t) = (x, u(x, t)) = (0, u t ) t T (x, t) = cu t (x, t) (cu t u x ) x = ρu tt Even from the resolution of the transport equation we realize that in orer to obtain only one solution to an equation we nee to impose extra conitions. For example, at a certain time it shoul take a certain form or on a certain omain it shoul be of a certain form. The most common assumptions, which are justifie by the original physical meaning of the PE, are initial conitions an bounary conitions. Initial Conitions: If we consier a transport equation it is reasonable to assign how the wave looks at time t = 0, so we write u(0, x) = g(x). We can o the same for the Schröinger equation. For the wave equation, just specifying u at t = 0 is not enough, we also nee the velocity at t = 0 so we a u (0, x) = f(x) Bounary Conitions: For the heat equation (stationary or not) it is important to specify how the solution looks like at, where is the boy. There are three well known bounary conitions: u - irichlet bounary conitions u - Neumann bounary conitions n u n + au - Robin conition where a is a given function in x, y, z an n is the unit normal vector to. efinition: If the bounary conition is constantly equal to zero, then the conition is sai to be homogeneous. If not, it is inhomogeneous. 6