18.311-MIT. Principles of Applied Mathematics. Spring 2012. Tu and Th 11:00--12:30. Room 2-190 Rodolfo R. Rosales. Lecture Summaries. % ========================================================================== % ---------------------------------------------------- Lecture 01 Tue-Feb-07 General mechanics of class. Discuss syllabus, grading, books, notes, etc. Outline of the mathematical topics to be covered. Physical phenomena that motivate the material: --- Hydraulic Jumps [kitchen; river floods; flash floods; dams; etc]. --- Shock waves [sonic boom, explosions, super-novas and crab nebula]. --- Traffic flow waves. Others (not mentioned in this lecture) --- Solitary waves [say, in lakes]. --- Diffusion. % ----------------------------------------------------------------------- % % --------- NOTE: for the material we start covering now, see ----------- % % --------- the conservation law notes on the WEB page. ----------- % % ----------------------------------------------------------------------- % % -------------------------------------------------------------------- #001 Derive the Lighthill-Whithams model for Traffic Flow rho_t + q_x = 0, where q is a function of rho. Brief discussion of the continuum limit involved: --- What is rho and under what assumptions one can define it. --- What is q and under what assumptions one can define it. --- Point out analogy with continuum hypothesis in fluids and solid matter. The difference between 10^20 and 10^1.5. --- Conservation of cars: Integral and differential forms. % % -------------------------------------------------------------------- #002 Closure problem: more unknows than equations. --- Quasi-equillibrium approximation leading to q = Q(rho). Assumption: there is a stable equillibrium to which system adjusts on a time scale faster than the time scales of interest in problem. --- Discuss generic form of Q. Existence of a maximum throughput q_m = Q(rho_m). % % ========================================================================== % ---------------------------------------------------- Lecture 02 Thu-Feb-09 % % -------------------------------------------------------------------- #003 Lighthill-Whithams model for Traffic Flow. Recall derivation. The flux function q = Q(rho): --- Discuss practical significance of the existence of q_m = Q(rho_m). Leads to entrance lights at highway access at peak time. Explain purpose, and limitations (no dynamics and not enough space to have cars wait till optimal rho_m is reached). % % -------------------------------------------------------------------- #004 GENERAL FRAMEWORK. Discrete to continuum limits. --- Densities and fluxes. --- Issues relating to the meaning of the "continuum" limit. Examples: Car densities and flux. Interstellar media (where do super-nova shocks travel on?). River density and flux. (see #005) Pressure in a gas. Other fluid properties. (see #006) Other examples: Grade "curve" (idealization versus reality: histograms). % % SKIP Forestry, molds, bacteria, ... --- examples later. % % -------------------------------------------------------------------- #005 EXAMPLE: River flow equations. A_t + Q_x = 0. Describe quasi-equillibrium function Q = Q(A). dQ/dA is increasses with A. Justify: at equillibrium forces (friction and gravity) must balance. --- Note that dq/drho (traffic flow) and dQ/dA (river floods) have the dimensions of a velocity. But, the velocity of what? We will answer this question soon. --- Compare q = Q(rho) and q = Q(A) for traffic flow and river flow. Concave versus convex. In one case c = dq/drho decreases and in the other increases. % % -------------------------------------------------------------------- #006 EXAMPLE: Euler Equations of Gas Dynamics in 1-D. Closure, quasi-equillibrium and thermodynamics. Polytropic gas. Body forces (sources). % % SKIP 2-D and 3-D. Compressible and incompressible cases. % % -------------------------------------------------------------------- #007 GENERAL FRAMEWORK. How conservation principles lead to pde's: --- Integral and differential form of a conservation law. --- Closure issues and "constitutive equations." Quasi-equillibrium approximations and time-scale limitations. The case of traffic flow and the traffic flow curve. Further examples (below) will clarify these ideas. Also, check the conservation law notes on the WEB page. % % -------------------------------------------------------------------- #008 EXAMPLE: Heat flow along an insulated wire. Describe closure, Fick's law, and do intuitive justification using stat. mech. interpretation of heat and temperature. % % -------------------------------------------------------------------- #009 More than 1-D. Conservation laws in 2-D or 3-D. Flux q is now a vector. Use Gauss theorem to obtain general form rho_t + div(q) = S (S equal sources & sinks). Examples of S: Cars flowing in/out of highway through commuter township. Water flowing into river from small affluents. Heating by electromagnetic waves [microwave oven]. % % ========================================================================== % ---------------------------------------------------- Lecture 03 Tue-Feb-14 Recap #008 [add intuitive justification of Fick's law] Recap #009 [add sources & sinks] % -------------------------------------------------------------------- #010 EXAMPLE: Heat flow in 2-D or 3-D. Then rho = r c_v T = conserved stuff (heat) per unit mass where c_v = specific heat of material, r = mass density T = temperature, Fick's law applies q = - kappa gradient(T), and kappa = heat conductivity. Thus: Heat equation: T_t = nu Laplacian T, where nu = kappa/(r c_v) = coeff. thermal diffusion. % % -------------------------------------------------------------------- #011 EXAMPLE: Diffusion equation (Salt in water, sugar in coffee, ink in water, ...) Same as heat equation C_t = nu*Laplacian C where C = concentration (salt, ink, sugar, ...) nu = difussion coefficient. % % -------------------------------------------------------------------- #012 DIMENSIONS and DIMENSIONAL ANALYSIS. What are the dimensions of kappa? nu? c_v? How long does it take sugar to sweeten a coffee cup without stirring? Idealized problem: start with a very small blob of ink, and ask: What is the radius of the blob of ink, R = R(t), as the blob expands due to diffusion? Dimensional analysis says R(t) \propto sqrt{nu*t} In particular, let L be the size of the coffee cup. Sugar will reach whole cup when R = O(L) ===> time = O(L^2/nu). Also the relevant time needed to cool/heat a size L vessel. These times are very long when measured in human-relevant scales. Hence stirring needed. Boiling and convection speed up heating. Questions: why does stirring help? why does convection occur? what would happen when heating something with a flame in the absence of gravity? At room temperature, in cm^2/sec Thermal diffusivity: water ~ 0.0014 mercury ~ 0.042 Diffusion in water: NaCl ~ 10^{-5} % % -------------------------------------------------------------------- #013 Conservation laws and pde. Higher order (TRANSPORT) effects beyond quasi-equillibrium. Note difference between heat/diffusion and the other examples. In one case the fluxes are functions of the densities only, in the other derivatives of the densities appear. --- Important when gradients are not so small. --- Connection with randomness at microscopic level. % % -------------------------------------------------------------------- #014 EXAMPLE. Another example of a higher order transport effect is viscosity: forces proportional to the flow velocity gradient. REASON: the flow velocity is a macroscopic "average" variable. The same phenomena that causes heat conduction for thermal energy, produces momentum transfer (forces), as well as kinetic energy transfer. A small peek at the Navier-Stokes equations: to the conservation-law derivation of the Euler equations (#006), add heat conductivity and viscosity. Derive compressible Navier Stokes in 1-D. We will/may come back to this later. % % -------------------------------------------------------------------- #015 % EXAMPLE: Slow granular flow in a silo. SKIP: Students should read this in the WEB page notes. % % ========================================================================== % ---------------------------------------------------- Lecture 04 Thu-Feb-16 % -------------------------------------------------------------------- #016 Conservation laws in more than 1-D. Meaning of the flow vector. Why is it a vector? For some conserved quantity, define the flow q as follows: q is the amount of rho per unit time and per unit length in 2D (area in 3-D) crossing a curve in 2D (surface in 3-D) from one side to the other. With this definition q is a (scalar) function that depends on the position and the orientation (normal) of the differentials of -- length for the curves across which the flux occurs (2D). -- area for the surfaces across which the flux ocurrs (3D). Then show that q must have the form q = \hat{n} \cdot \vec{q}, where \vec{q} is a vector, and \hat{n} is the unit normal to the curve/surface. [then q positive means that the net flow is in the direction of \hat{n}]. Argue using the standard rectangular simplex of "infinitesimal" size (limit is involved), and conservation on an area (volume) that vanishes much faster than the fluxes through the sides --- which, hence, must balance. % % ------------------------------------------------------------------------- NEXT TOPIC: Solution of 1-st order (scalar) quasilinear equations by characteristics. Examples: traffic flow and river waves. This will take about 4 lectures. % ----------------------------------------------------------------------- % % --------- NOTE: for the material we start covering now, see ----------- % % --------- the books by Wan, Haberman, Whitham. ----------- % % ----------------------------------------------------------------------- % % ======================================================================= % % Review of chain rule in several variables. % It is important that the students be familiar with differentiation % % with several variables, chain rule, and implicit differentiation. % % Please review your 18.02 notes. % % ======================================================================= % % -------------------------------------------------------------------- #017 Traffic Flow equations. Traffic density, flow and car velocity. Linearize equation near a constant density and solve u_t+c_0*u_x = 0. Do it first by separation of variables/Fourier. Show solution has form u = f(x - c_0*t). This works because equation is linear and constant coefficients. Need approach that will generalize to u_t + c(u)*u_x = 0. But first, note that: for traffic flow u > c and for river flows u < c. Density waves reach cars from ahead. River waves move faster than the flow. Interpret equation as a statement about a directional derivative of the solution in space-time being zero. Leads to the solution being constant along the integral curves for the directional derivatives. Definition of characteristics and solution by characteristics. Applies to any equation that can be translated into statements about the directional derivatives of the solutions. Such equations are called hyperbolic. Scalar hyperbolic equations can be reduced to o.d.e.'s along the curves integrating the corresponding directional derivatives. % % -------------------------------------------------------------------- #018 Examples: Linear problems with constant or simple variable coefficients, where all the calculations can be done exactly. In each example: --- Write characteristics in parametric form. --- Solve and draw the characteristics. --- Eliminate the characteristic variables and find the solution. --- Show where the solution is defined. Example 1: u_t+c_0*u_x = a*u. IVP problem on -inf < x < inf, t > 0. Example 2: x*u_x + y*u_y = y, u(x, 1) = g(x) for -inf < x < inf, y > 0. Example 3: u_x + x^2*u_y = y, with u(x, 0) = g(x) for x > 0. Show that this defines the solution to the right of y = x^3/3. % % ========================================================================== % ---------------------------------------------------- Lecture 05 Thu-Feb-23 Finish #018 [do examples 2 and 3]. Chain rule in several variables. This was done before. Remind students to review this topic. % -------------------------------------------------------------------- #019 Describe general method, for a problem of the form a(x, y)*u_x + b(x, y)*u_y = c(x, y), u given along some curve. That is: u = U(z) on some curve x = X(z) and y = Y(z). Equations: dx/ds = a, dy/ds = b, and du/ds = c, to be solved with the conditions x = X(z), y = Y(z) and u = U(z) for s = 0. Leads to solution expressed in the form u = u(s, z), with x = x(s, z) and y = y(s, z). where: z = parameter/label for the characteristic curves. s = parameter that results from solving o.d.e.'s above = parameter along each characteristic curve. (s, z) is a curvilinear, coordinate system. Characteristic coordinates. To get the solution must solve for s and z as functions of x and y. Illustrate the coordinates (s, z) graphically. % % -------------------------------------------------------------------- #020 In all the examples above, the characteristic curves are independent of the solution. This follows from the equations being LINEAR. That is a, b, and c do NOT depend on u. % % ------------------------------------------------------------------------- % ----------------------------------------------------- % % ----------------------------------------------------- % % ----- % Next we move up to NONLINEAR PROBLEMS % ----- % % ----------------------------------------------------- % % ----------------------------------------------------- % % -------------------------------------------------------------------- #021 Example 4: General kinematic wave equation u_t + c(u)*u_x = 0, with u(x, 0) = f(x). -1- Characteristic form and characteristic speed. -2- Solution generally cannot be written explicitly. -3- Geometrical interpretation of the solution. Leads to a clear picture of how conservation is achieved: "SLIDING SLABS" image. -4- Wave distortion/steepening and wave breaking. -5- Smooth solution do not exist for all time. -6- Show that the characteristics can/do cross in space time. -7- Derivatives become infinity at time of first crossing. NOTE: dc/du < 0 for traffic flow, and dc/du > 0 for river flows. Consequences for wave steepening: waves steepen backwards (TF) or forwards (RF). Matches observations % % ========================================================================== % ---------------------------------------------------- Lecture 06 Tue-Feb-28 Finish #021. Did only -1- in prior lecture % % -------------------------------------------------------------------- #022 Back to nonlinear kinematic waves and wave-breaking: After wave breaking, need more physics to fix the problem. Explain: why does the model fail there? (quasi-equilibrium). Fix to the problem is NOT in the mathematics. Need NEW physics! Will study traffic flow to learn how to fix it. % % -------------------------------------------------------------------- #023 Back to traffic flow, and example problems there. --- Red light turns green. Gap in characteristic field. Argue nice dependence on perturbations to data. Fill gap by taking limit of smeared discontinuity. Obtain expansion fan solution c = x/t. Explicitly solve for rho in the case: q = q(rho) is quadratic. Alternative: use the "sliding-slab evolution" picture (#021) to get the solution Physical meaning of the various waves observed: --- Path of first in line behind the red light. A characteristic, and also a car path, since u(0) = c(0). --- Location where cars start moving behind the light. Light wave: locus where brake lights turn off. --- Green light turns red. Split into two initial-boundary value problems (ahead and behind the light). We will do both problems, and clarify the physical meaning of the various waves observed. Do case of "light" traffic first. Ahead of light: characteristics cross. Resolve issue by introduction of the last car to make the light. Discontinuity in solution where characteristics are chopped. Speed of this discontinuity is obvious (path of last car through the light). Introduce notion of curves along which characteristics die as a solution to the problem of multiple values. This works in this case, at least. We'll see it is generic. Notice that this is new math. into the model. % ========================================================= % % % % LECTURE ENDED % % % % ========================================================= % Note that, since the discontinuity here is at the last car through the light, conservation follows trivially. Below we will see what is the general rule for the speed of the places where characteristics die [discontinuities/shocks]. We will also see that the general rule below reduces to: Shock speed = car speed when state behind is rho = 0. Behind the light: again, characteristics cross. Argue that in real life drivers wait till the last moment to break. Another discontinuity needed: Location of thin layer where cars break. Wave moving backwards from the light. How does one compute it's velocity? Get law speed = [Q]/[\rho] by arguing "flow of cars into discont. = flow of cars out" Note flow = rho*(u-s). Note limit as [rho] vanishes is characteristic speed. Again: characteristics chopped at discontinuity and crossing avoided. Graphical interpretation of shock jump condition: ***** slope of secant line in rho-q=q(rho) diagram. ***** Note that, as shock jump vanishes, this becomes characteristic speed. % % ========================================================================== % ---------------------------------------------------- Lecture 07 Thu-Mar-01 FINISH #023. See mark for end of lecture in #23. % % -------------------------------------------------------------------- #024 RECAP of key points (see #022-23). Consider Traffic Flow with smooth initial wave profile rho = f(x). Or Flood waves """""""""""""""""""""""""""""""""""""""""""" List resulting differences in behavior. % % -------------------------------------------------------------------- #025 Characteristics almost always cross. WRITE THE PRECISE CONDITIONS NEEDED FOR THIS TO HAPPEN. Simple problem: rho_t + q_x = rho_t + c(rho)*rho_x = 0, rho(x, 0) = R(x), c(rho) = dq/drho. Solution by characteristics: x = X(s, t) = C(s)*t + s, (#1) rho = R(s), where C(x) = c(R(x)) = wave speed along initial data. Characteristics do not cross if and only if can solve for s as a function of x and t --- s = S(x, t) --- from (#1) if and only if map s --> x is monotone: X_s not 0. That is: inspect X_s = C'(s)*t + 1. So, if C'(s) < 0 somewhere, there will be a time when X_s = 0. Thus, the condition for crossing is: dc/dx < 0 somewhere in the initial data. Graphics: show how x = X(s, t) = C(s)*t+s looks like as a function of s, for t fixed, as t grows, if C(s) is a localized hump. t = 0 straight line x = s. t >0 moderate straight line develops a wiggle. t >0 large wiggle large enough to produce a local max. and a local min. Hence a range where map is not 1-to-1. Formula for critical time t_c and location x_c where the characteristics cross first, assumming C'(s) < 0 somewhere (no crossings otherwise) Let s_c be the value of s at which C'(s) reaches its largest negative value (i.e.: absolute minimum). Then t_c = - 1/C'(s_c) ... so X_s vanishes. x_c = C(s_c)*t_c + s_c % % -------------------------------------------------------------------- #026 Evolution of wave profile, as given by the characteristic solution. Graphical interpretation: -- Move each point on graph at velocity c(rho). Evolution as sliding of horizontal slabs at different velocities (guarantees conservation). -- Causes wave steepening, wave breakdown, and multiple values. -- Mathematical model breakdown. Quasi-equillibrium assumption fails and PDE model breaks down. -- Back to "physics". Need to augment model with new physics, namely: shocks in the case of traffic flow or flood waves. % % -------------------------------------------------------------------- #027 First look at EXPANSION FANS: A discontinuity in the initial, or boundary, conditions gives rise to an expansion fan if the edge characteristics do not cross and leave a gap. That is: c is increasing across the discontinuity: EXAMPLES: Red-light turns green. Red-green-red-green ... Will see more later. DEFINITION: An expansion fan is the solution produced by a collection of characteristics, all starting at one single point, but with a range of values for the solution there. % % -------------------------------------------------------------------- #028 First look at SHOCKS: A discontinuity in the initial, or boundary, conditions gives rise to shock if the edge characteristics cross. That is: c is increasing across the discontinuity: EXAMPLE: green-light turns red problem. One shock on each side of light. Shock on the right: last car through light. Shock on the left: locus where cars break behind the light. Shock conditions (see #027). Rankine-Hugoniot shock speed S = [q]/[rho]. Derive from conservation: In shock frame: flow from the left = rho_*(u_ - S) = F_ In shock frame: flow from the right = rho+*(u+ - S) = F+ Then conservation is F_ = F+, which gives S = [q]/[rho]. Graphical interpretation of Rankine-Hugoniot: slope of secant line in rho-q diagram. Note: limit for infinitesimal strength shocks is characteristic speed. % % -------------------------------------------------------------------- #029 Important: crucial to know what is conserved. Note that the equation u_t + u*u_x = 0 can arise from various conservation laws: (a) Conserved density rho = u, flux q = (1/2)*u^2, (b) Conserved density rho = u^2, flux q = (2/3)*u^3 (c) Conserved density rho = u^3, flux q = (3/4)*u^4 .... Each of this gives rise to the same solution as long as there are no shocks, but give rise to DIFFERENT shock speeds, hence different solutions once shocks arise. % % ========================================================================== % ---------------------------------------------------- Lecture 08 Tue-Mar-06 % % -------------------------------------------------------------------- #030 Summary of traffic flow: what is conserved, shocks, RHJC, Get Burger's equation after a change of variables u = q' = c (wave speed) ==> u_t + uu_x = 0 Difference in conserving rho versus u -- different shock speeds % % -------------------------------------------------------------------- #031 Examples of shocks for Burger's (if conserve u) -- u(x,0) = 1 (x<0), 0 (x>0) -- u(x,0) = 0 (x<0), ax (0b) -- u(x,0) = u_1 (x<0), u_1 (a-x)/a (0=a) -- u(x,0) = 0 (x<0), b*x/a (0a) % % ========================================================================== % ---------------------------------------------------- Lecture 09 Thu-Mar-08 % % -------------------------------------------------------------------- #032 Lax's Entropy condition // stability of the shocks u_L > shock speed s > u_R, where u_L = char speed to the left, and u_R - to the right of the shock Lax's definition of the shock: a shock is a discontinuity that satisfies RH jump condition and the entropy condition Graphics: given q(rho) for the traffic flow and points rho_L and rho_r, 1) what are characteristic speeds at rho_l and rho_r graphically 2) in which case there will be a shock forming (investigate rho_L < rho_r, and rho_l>rho_r). 3) speed of the shock graphically is the slope of the secant line % % -------------------------------------------------------------------- #033 Envelopes Definition, system of equations to solve to get the envelope If a one-parameter family of curves is given by F(x,y,s)=0, then to find the envelope of these curves one has to solve a system of equations F(x,y,s) = 0, F_s(x,y,s) = 0 Two examples: (1) x/s + y/(1-s) = 1 and wave front from a parabolic disturbance (if wave speed = const) (2) (x-x_0)^2 + (y-x_0^2)^2 = R^2 (= const) % -------------------------------------------------------------------- #034 Envelope of characteristics: The boundary of the region of multiple values is the envelope of the characteristics. % % ========================================================================== % ---------------------------------------------------- Lecture 10 Tue-Mar-13 % % -------------------------------------------------------------------- #035 Motion of surfaces: Shocks in n-dimensions are surfaces of discontinuity Derive the equation for the motion of surfaces Eikonal equation Special case of V = const gives "signed distance function". Physically: Huygen's principle Get a *nonlinear* first order PDE u_x^2 + u_y^2 = 1/V^2 = 1 (when V = const = 1) To solve it, introduce new variables and convert the equation to a first order quasi-linear PDE H(X,Q,u) = 0, where X = (x,y), Q = (u_x, u_y) Solve by the method of characteristics (general approach). EXAMPLE: Redo the problem with the initial data on (x0,y0) = (x0, x0^2). Matlab example: solution to Eikonal equation, start with any profile. Show an example of an initial profile concave outward, which gives cusps in finite time -> caustics. % % ========================================================================== % ---------------------------------------------------- Lecture 11 Thu-Mar-15 % % -------------------------------------------------------------------- #036 Quasi-linear first order PDEs (QL PDEs) Are equations of the type a(x,y,u)u_x + b(x,y,u) u_y = c(x,y,u). (1) Find: u(x,y), given initial data on some curve: x = f(s), y = g(s), u = h(s). (2) Solve by the method of characteristics: Introduce a system of ODEs on characteristics dx/dt = a, dy/dt = b, du/dt = c (3) Solve it phi_1 (x,y,u) = c_1 (s) (4) phi_2 (x,y,u) = c_2 (s) Then exclude s and get that phi_1 is some unknown function of phi_2: phi_1(x,y,u) = F ( phi_2 (x,y,u) ) (5) To find this function F, plug in the initial data (2) into (5). Then (5) becomes a solution to (1). It might not have u(x,y) in an explicit form, but that's life. % % ========================================================================== % ---------------------------------------------------- Lecture 12 Tue-Mar-20 % % -------------------------------------------------------------------- #037 Burger's equation: u_t + uu_x = nu u_xx The simplest equation that has nonlinear advection and diffusion. We show first, that one can use Cole-Hopf transformation, which is a *nonlinear* change of variables, to convert the Burger's equation into a *linear* heat equation. The change of variables (chovar) is: u = - 2*nu*phi_x / phi . Note: If you want to derive the same way we did in lecture, you can use first the chovar u = psi_x, and then 2nd chovar psi = - 2*nu*log(phi). The heat equation will be in the variable phi: phi_t = nu*phi_xx. % % -------------------------------------------------------------------- #038 Derivation of the solution of the heat equation: u_t = nu u_xx Physical application: diffusion of heat, T = temperature There are several ways to approach the solution to the heat equation. (I) separation of variables u(x,t) = f(x)g(t), plug in, solve ODEs for g(t) and f(x), from physics require that g(t) is a decaying exponential, deduce that f(x) is sum of sines and cosines. Solve an IBVP (initial-boundary-value-problem) on [0,L] u(x,0) = u_0(x), u(x = 0, t) = u(x = L,t) = 0. (II) Green's function --- Start with dimensional analysis: the only non-dimensional group is eta = x / sqrt(nu t). --- Assume that the solution is dependent on the function of eta, u(x,t) ~ (proportional to) ~ f(eta) --- since integral of u(x,t) dx over the real line is constant, say M_0, which we derived form the heat equation, deduce that u(x,t) = M_0 / sqrt(nu t) f(eta) --- (next lecture) Find f(eta) by plugging in the expression above into the heat equation. Normalize f(eta), so that its integral over the real line is 1. Observe that you got the Green's function. % % ========================================================================== % ---------------------------------------------------- Lecture 13 Thu-Mar-22 % Continue with #038 % Derive the fundamental solution to the heat equation u_t = nu u_xx (II) Green's function --- Start with dimensional analysis: the only non-dimensional group is eta = x / sqrt(nu t). --- Assume that the solution is dependent on the function of eta, u(x,t) ~ (proportional to) ~ f(eta) --- since integral of u(x,t) dx over the real line is constant, say M_0, which we derived form the heat equation, deduce that u(x,t) = M_0 / sqrt(nu t) f(eta) --- Find f(eta) by plugging in the expression above into the heat equation. Solve the ode of f(eta): f'' + 1/2 (f eta)' = 0 obtain f(eta) = C*exp( - eta^2 /4 ). Normalize f(eta), so that its integral over the real line is 1. --- Aside: integral_(-infty)^(infty) exp( - x^2 ) dx = sqrt(pi) (***) --- Normalized f(eta) is f(eta) = 1/sqrt(4 pi) exp( - eta^2 /4 ). --- and the solution to the heat equation is: u(x,t) = 1/sqrt(4 pi nu t) exp( - x^2 / (4 nu t) ) --- Notice, that as t->0 u(x,t)-> delta(x), i.e. a delta function centered at x = 0. To center it at an arbitrary location, introduce Green's function G(x-x0,t) = 1/sqrt(4 pi nu t) exp( - eta^2 / 4 ) with properties: (1) it solves the heat equation with initial data delta(x-x0) (2) the integral over the real line is 1 (3) since the heat equation is linear, given any initial data u0(x), the solution to the heat equation is u(x,t) = int_(-infty)^(+infty) G(x-x0,t) u0(x) dx % % -------------------------------------------------------------------- #039 Laplace's method % --------------------- Motivation: --- Question: For the Burger's equation u_t + uu_x = nu u_xx what happens when nu -> 0 ??? Plan: after Cole-Hopf transformation, u = - 2 nu phi_x / phi, phi solves the heat equation phi_t = nu phi_xx. We know the solution to that. So take the limit nu -> 0 there. --- Question: How? -- Asymptotic expansion of integrals Laplace's method: gives the leading order behavior of the integrals of the type int_a^b f(x) exp( lambda S(x) ) dx for lambda -> infty. % --------------------- Steps: --- if S(x) has a unique max on a infty): int_a^b f(x) exp( lambda S(x) ) dx ~ f(x0) exp(lambda S(x0)) sqrt( -2 pi / (lambda S''(x0)) ) % % ========================================================================== % % spring break Mar 26 -- Apr 1st % % ========================================================================== % % ========================================================================== % ---------------------------------------------------- Lecture 14 Tue-Apr-3 % % -------------------------------------------------------------------- #040 Finish Burger's equation: Summary: --- to see if the shock solutions are physical, we consider a more complicated model that includes more physics -- the Burger's equation. u_t + uu_x = nu u_xx. --- We find the exact solution to it via the trick of Cole-Hopf transformation, that transforms Burger's into the Heat equation for the new variable phi: phi_t = nu phi_xx for which we know the exact solution. --- Transforming the solution back to u, we have the exact solution to Burger's. --- We take the limit of viscosity nu->0 and use Laplace's method (#039) to find their asymptotic behavior. --- We recover the solution that we had using the method of characteristics: u(x,t) = u_0 (x_0) x = x_0 + u(x,t) t (in the lectures the notation used for x_0 was ski). --- For the case of two characteristics intersecting, we got equal-area relation for the shock. (next lecture) % % ========================================================================== % ---------------------------------------------------- Lecture 15 Thu-Apr-5 % % -------------------------------------------------------------------- #041 Whitham's equal area principle for shocks % % -------------------------------------------------------------------- #042 Examples of wave phenomena Example 1: Travelling wave solution Linear advection, homogeneous c_t + Vc_x = 0, V = const, c(x,t=0) = c_0(x) This equation can model a pollution in a channel (where the pollutant is uniform in the direction across the channel, and varies in the x direction along the channel). Solve by the method of characteristics x = x0 + Vt, c(x,t) = c_0(x_0). => travelling wave solution c(x,t) = c_0(x - Vt). Note: in some problems you want to look for a travelling wave solution, because of the physical properties of the system. For example, the Fisher-Kolmogoroff equation models the spread of a virus u_t = u(1-u) + nu u_xx, u = u(x,t). Here u is the population of the virus, so u = 0 stands for the unpolluted region with no virus, and u = 1 is the virus infested region. One interesting setup is what happens for the solution with u->1 at x->-infty, and u->0 at x->+infty. One can find a travelling wave solution of the form u(x,t) = f(x-Ut), U = const. This is a steady infection front propagating with constant speed U. Example 2: Decaying pollution source: c_t + Vc_x = - ac, V = const, a = const > 0. Linear non-homogeneous equation. Solve by the method of characteristics: dc/dt = - ac, dx/dt = V -> x = x0 + Vt, c = c_0 (x_0)exp(-at) Solution: c(x,t) = c_0(x-Vt) exp(-at) is a decaying travelling wave. Example 3: Damped nonlinear wave. If there is dissipation (example 2) and nonlinearity (e.g. Burger's), will the wave break or will the dissipation "win"? c_t + cc_x = - ac, c = c(x,t), c(x,0) = c_0(x), a = const > 0. Solve by the method of characteristics: dc/dt = - ac dx/dt = c -> c = c_0(x_0)exp(-at), dx/dt = c_0(x_0)exp(-at) The solution is: x = x_0 + c_0(x_0)(1-exp(-at))/a, c = c_0(x_0)exp(-at). When will there be shocks? When there is an envelope of solutions: 0 = 1 + c_0'(x_0)(1-exp(-at))/a. Since a>0 and t>0, this is only possible for steep enough initial profiles with negative enough initial slope: c_0'(x_0)<-a. % % ========================================================================== % ---------------------------------------------------- Lecture 16 Tue-Apr-10 % BEGIN NEW TOPIC % ======================================================================= % % %%%%%%%%%%%%%%%% Gas Dynamics, Acoustics, and Strings. %%%%%%%%%%%%%%%% % % ======================================================================= % % % -------------------------------------------------------------------- #043 Gas Dynamics: Derive equations in 1-D (using conservation of mass and momentum), plus the quasi-equilibrium isentropic assumption p = p(rho). Example: p = kappa*rho^gamma (polytropic gas) Polytropic gas: ideal gas with constant specific heat. % -------------------------------------------------- % % Review topic from thermodynamics for ideal gases: % % Show p = rho^gamma for ideal gas with constant % % specific heat at constant entropy. % % -------------------------------------------------- % --- Boundary conditions at the end of the pipe: Closed pipe u = 0; and Open pipe p = p_0; --- For smooth solutions, manipulate equations into the form rho_t + u*rho_x + rho*u_x = 0 u_t + (a^2/rho)*rho_x + u*u_x = 0 where a^2 = dp/drho. Note that dp/drho > 0, and that a is a velocity (sound speed, as we will see). Calculate a for polytropic gas case. % % -------------------------------------------------------------------- #044 Derive also Shallow Water equations on a flat bottom. --- Note: same equations as isentropic Gas Dynamics, with gamma = 2. --- Compute what a is for Shallow Water. --- Calculate a for tidal wave in the deep ocean (4.000 m). % % -------------------------------------------------------------------- #045 Characteristics for Gas Dynamics and Shallow Water waves. Note that the equations can be written in the vector form Y_t + A(Y)*Y_x = 0. (1) --- Similar to the scalar case, but with the wave velocity replaced by a matrix. --- Many other equations in applications can be written in this form: e.g.: Maxwell, Magneto-hydrodynamics, Reacting Gas Dynamics, etc. Now, look for o.d.e. forms [i.e. characteristics] by doing linear combinations of the equations. Need to find combinations that produce only one directional derivative in space-time. DEFINITION. Hyperbolic: equations that are EQUIVALENT to statements about directional derivatives of the solution. Show how to do this by using eigenvalues and eigenvectors of the matrix A (left eigenvectors) L*A = c*L. Note: left multiplying the equation by a row vector is equivalent to doing a linear combination of the equations. Characteristic form: L*(Y_t + c*Y_x) = 0, or L*dY/dt = 0 along dx/dt = c. Then, along the curves dx/dt = c, the solution behaves (sort of) like an o.d.e. Hyperbolic then means: have enough linearly independent (real) eigenvectors (in Gas Dynamics or Shallow Water this means 2) so that equations are equivalent to a bunch of equations involving directional derivatives. This happens if and only if A is real diagonalizable. % % ========================================================================== % ---------------------------------------------------- Lecture 17 Thu-Apr-12 % % -------------------------------------------------------------------- #046 Example: consider a system of N equations of the form Y_t + A*Y_x = 0. (1) where A is a constant, NxN, real diagonalizable matrix, and use the ideas in #045 to write the general solution to (1). [See item #049] Note that general solution: --- Involves N arbitrary (scalar) functions. --- Each function corresponds to a signal propagating at speed lambda_n, where lambda_n is an eigenvalue of A. --- There is no wave distortion. The wave shapes propagate rigidly. % % -------------------------------------------------------------------- #047 Next we apply the ideas in #045-#046 to Gas Dynamics (and Shallow Water, which is the particular case gamma = 2). However, before applying these ideas to the full Gas Dynamics problem, we LINEARIZE near the equilibrium solution u = 0, and rho = rho_0, and analyze the resulting problem (this is ACOUSTICS). NEXT ITEM. % ====================================================================== % % The material below can be (partly) found in the book by Wan. % % ====================================================================== % % -------------------------------------------------------------------- #048 % ----------- Linear Gas Dynamics - Acoustic in a pipe ----------- % % -1- Write equations: R_t + rho_0*u_x = 0 and u_t + (a_0^2/rho_0)*R_x = 0, where R is the density perturbation to rho_0 [rho = rho_0 + R] Boundary conditions for closed (u = 0) and open (R = 0) pipe ends. -2- Find eigenvalues and eigenvectors c = +/- a_0 and L = (+/- a_0, rho_0) Write in characteristic form and solve. Solution superposition of two waves: left and right going sound waves. -3- Show equivalent to wave equation: Eliminate either R or u. Better yet, introduce velocity potential: u = phi_x and (a_0^2/rho_0)*R = - phi_t \--------------/ Note that this is the pressure perturbation. Hence, second equation is satisfied. Then first equation gives phi_tt - a_0^2*phi_xx = 0. Wave equation. Note boundary conditions: Closed pipe: phi_x = 0. Open pipe: phi = 0. -4- Use (2-3) to write general solution for the wave equation. % % -------------------------------------------------------------------- #049 Needed linear algebra facts (sketched proofs for most of them). Note: here we are only interested in real matrices, though the results have easy generalizations to complex matrices. a) A symmetric matrix has real eigenvalues, and the eigenvectors can be taken to be an orthonormal basis. b) If {n_j}, 1 <= j <= N, is an orthonormal basis of R^N, then any vector Y can be written in the form Y = sum_1^N y_j n_j, where y_j = , where < , > = scalar product. c) Let A be a real-diagonalizable square matrix. Let R_j, 1 <= j <= N, be a set of N linearly independent right (column) eigenvectors of A A R_j = lamda_j R_j, where the lambda's are the eigenvalues [they need not be distinct]. Then a set of N linearly left (row) eigenvectors of A L_j A = lambda_j L_j can be selected such that L_n^T R_m = delta_{n, m}, where delta_{n, m} is the Kronecker delta. Then any (column) vector Y can be written in the form Y = sum_1^N y_j R_j, where y_j = L_j^T Y. This formula generalizes the one in (a-b), which applies for symmetric matrices. % % ========================================================================== % % April 17th -- Patriot's day -- no class % % ========================================================================== % % ========================================================================== % ---------------------------------------------------- Lecture 18 Thu-Apr-19 % % -------------------------------------------------------------------- #050 Derive characteristics for the full 1-D isentropic Gas Dynamics. rho_t + u*rho_x + rho*u_x = 0 u_t + (a^2/rho)*rho_x + u*u_x = 0 where a^2 = dp/drho. Write in the form Y_t + A(Y)*Y_x = 0, and find the eigenvalues and left eigenvectors for A (namely: solve L*A = c*L). Then c = u +/- a, and the characteristic form is: +/- (a/rho)*(drho/dt) + du/dt = 0 along dx/dt = u +/- a. Two sets of characteristics, which interact and couple. The situation similar to u_tt - u_xx + V(u) = 0 [we will see this example later, in #065], but more complicated: Here the characteristic speed is not constant. The characteristics in the same family may cross ... leading to shocks. Introduce h = h(rho) by property dh/drho = a/rho. Show for ideal gas h = 2*a/(gamma-1). Then d/dt (u +/- h) = 0 along dx/dt = u +/- a. i.e (u +/- h) is constant along characteristics. Show how this, in principle, determines the solution. At each point in space time two characteristics [C+ and C-], each carrying information from a different part of the initial data, which combined gives the solution at the point. But now, unlike the case of traffic flow, the characteristics are neither straight, nor can we solve for them explicitly (because they interact with each other). % % -------------------------------------------------------------------- #051 Example: simple waves in Gas Dynamics. Assume u-h = constant for initial data. Then characteristics yield u-h = L = constant, as long as characteristic form applies. Hence u = h + L = U(rho) is a function of rho only. The equations then reduces to d/dt (u+h) = 0 along dx/dt = u+a. That is (U+h)_t + (u+a)*(U+h)_x = 0, which is a first order equation of the same type as Traffic Flow and River Flows. Hence characteristics can cross and once this happens we need to re-examine the physics to see what to do beyond breakdown. In this case, again, shocks are the appropriate answer. % ======================================================================= % % % % Below: skip from #052 up to, and including, #058. % % % % ======================================================================= % % -------------------------------------------------------------------- #052 % SKIP Shocks in Gas Dynamics: Rankine-Hugoniot and Entropy conditions. % % -------------------------------------------------------------------- #053 % SKIP Piston problem in Gas Dynamics. Use Riemann invariant form of equations in simple problem: -- Gas in a pipe, initially at rest: rho = rho_0 and u = 0, x > 0. -- Some signal at x = 0, t > 0 (sound) u = sigma(t) -- Argue left Riemann invariant constant through-out flow, provided no shocks. Reduce problem to finding right R. I. -- Point out problem is now very similar to Traffic flow. -- Do case sigma(t) = V < 0 (constant). Show need for expansion fan in C+ characteristics -- Do case sigma(t) = V > 0 (constant). Show C+ characteristics cross. Need shock, and left Riemann invariant assumption has to be re-examined. % % -------------------------------------------------------------------- #054 % SKIP (optional material). The p-system --- Characteristic form. Riemann invariants and SIMPLE waves. Shocks. --- Isentropic Gas Dynamics in Lagrangian Coordinates is the p-system (change of variables). --- p-system same as equation for a nonlinear vibrating elastic rod. % % -------------------------------------------------------------------- #055 % SKIP (optional material). Weak shocks in Kinematic Waves. --- Weak shock velocity (average of characteristic speeds). --- Weak shocks and quadratic approximation for flux. % % -------------------------------------------------------------------- #056 % SKIP (optional material). General problem of shock formation in Kinematic Waves. -- Shock formation time and envelope of characteristics. -- Envelopes. Examples of envelopes and shocks. % % -------------------------------------------------------------------- #057 % SKIP (optional material). % This was covered by Lyuba. See #037 -- #040 Burgers' equation. -- Cole-Hopf transformation: exact solution. -- Solution of heat equation by Green's function and application to Burgers. -- The small viscosity limit in Burgers'. Laplace's method. % % -------------------------------------------------------------------- #058 % SKIP (optional material). Toy model for sonic booms [pdf notes for web page being written]. Full problem requires 3-D and motion of finite body through air. -- Very messy. Simplify dramatically, to bare bone effects. Use axe if needed: -- Not interested in details near the plane, so think of the plane as a point source of momentum. -- Knock the problem to 1-D. But then loose the fact that in 3-D the waves, as they move away from the plane, decay [geometrical effect due to expansion]. Replace this by adding some ``damping'' into the system. -- Replace Gas Dynamics by the simplest (nonlinear) model with wave velocity dependence on the solution, e.g. use a simple wave model. Thus, the TOY MODEL is p_t + ((1/2)*p^2)_x = delta(x-V*t) - a*(p-p_0), a > 0. where p_0 = "ambient" state into which plane moves. Now -- Use a Galilean transformation to set p = p_0 -- Explain meaning of delta function (point source of p) and derive jump conditions across source path x = V*t. -- Write characteristic equations for problem away from x = V*t. Assume initial conditions p = p_0 = 0, and solve problem: Case V < 0. -1- Solution behind plane trivial. -2- Show characteristics starting behind the plane reach it, go through it, and determine solution ahead of plane. These do not cross, and set-up a steady state ahead of plane (show solution by characteristics yields p = p(x-V*t) ahead of plane. -3- There is a shock separating -2- from the air at rest far ahead of the plane. But this shock has strength that vanishes as it moves further and further away from plane. -4- Solution is smooth steady state in t --> infinity limit. Case V > 0 (but not too large) -5- Show need expansion fan behind the plane. -6- Steady state solution analog to 2 has characteristic crossing and needs a shock. -7- Solve for steady state and show solution has a square root singularity. -8- Steady state, t ---> infinity solution, now has shock riding ahead of plane. Case V > 0 "large" (plane catches up to shock ahead). % ----------------------------------------------------------------------- % % % % >>>>>>>>>>>>>>> Still to write. <<<<<<<<<<<<<<<<<<<< % % % % ----------------------------------------------------------------------- % % % -------------------------------------------------------------------- #059 Characteristic form for wave equation: u_tt - c^2*u_xx = 0. WAY 1. Use the relationship with acoustics and its characteristics. This is what we did in #048. Equivalently: Transform equation into 1st order system introducing a potential. -- Equation is curl(V) = 0 for vector field V = [u_t, c^2 u_x]. -- Remind/prove theorem: Irrotational vector fields in simply connected domains are gradients. -- Remind students to review the Green/Gauss/Stokes theorems in calculus. -- HENCE u_t + \phi_x = 0 and \phi_t + c^2 u_x = 0, which is acoustics with phi = delta p = (a_0^2/rho_0)*R and c = a_0. REVIEW of prior material [#048]: A) Linearized gas dynamics: rho_t + rho_0*u_x = 0 and u_t + (c_0^2/rho_0)*rho_x = 0. B) Show that (A) is equivalent to the wave equation (eliminate either rho or u). C) Write (A) in characteristic form and solve: C+) u_t + c_0*u_x + (c_0/rho_0)*(rho_t + c_0*rho_x) = 0 ===> rho_0*u + c_0*rho = constant on the characteristic x = c_0*t + x_0 C-) u_t - c_0*u_x - (c_0/rho_0)*(rho_t - c_0*rho_x) = 0 ===> rho_0*u - c_0*rho = constant on the characteristic x = -c_0*t + x_0 Thus the solution is a superposition of two waves. % % -------------------------------------------------------------------- #060 Characteristic form for wave equation: u_tt - c^2*u_xx = 0. WAY 2. This approach is more symmetric, and independent of any "special" connections of the wave equation with other equations/systems. In fact, it goes back to the basic idea behind characteristics: Look for linear combinations of u_t and u_x that yield equations with directional derivatives only. This gives: (u_t + c*u_x)_t - c*(u_t + c*u_x)_x = 0 (u_t - c*u_x)_t + c*(u_t - c*u_x)_x = 0 Solve this and get again solution. % % ========================================================================== % ---------------------------------------------------- Lecture 19 Tue-Apr-24 % % -------------------------------------------------------------------- #061 Characteristic form for wave equation: u_tt - u_xx = 0. WAY 3. Here we set c = 1 to simplify notation. #060 suggests the following approach: Introduce the variable v = u_t - u_x, and write the equation in terms of it. The resulting equation is a first order linear equation. Solve this equation (find its general solution). Again: get that the solution to the wave equation consists of two waves, moving right and left at speed c = 1. % % -------------------------------------------------------------------- #062 Characteristic form for wave equation: u_tt - u_xx = 0. WAY 4. Here we set c = 1 to simplify notation. An approach more "symmetric" than the one in #061 is to write the equation in terms of the two variables v = u_t - u_x w = u_t + u_x This will prove useful when we deal with a more general "wave equation" below in #065. That is: u_tt - u_xx + V(u) = 0. % % -------------------------------------------------------------------- #063 Waves on strings Derive wave equation for small transversal motion of an homogeneous string under tension, using conservation of transversal momentum. (rho*u_t)_t - (T*u_x)_x = 0. rho = mass per unit length .... Thus rho*u_t = momentum density. T = tension ................. Show how -T*u_x is the momentum flux (force from xa). Hence: u_tt - c^2*u_xx = 0, where c = sqrt(T/rho). Check units are correct. If string tied at ends: u vanishes there. If string loose at ends: u_x vanishes there. % % -------------------------------------------------------------------- #064 String on an elastic bed. This adds a restoring force at each point. Equation becomes u_tt - c^2*u_xx + b*u = 0, where b = k/rho and k is the elastic restoring force constant. A more general restoring force yields u_tt - u_xx + V(u) = 0. % % -------------------------------------------------------------------- #065 Characteristics for the equation: u_tt - c^2*u_xx + V(u) = 0. This equation can be written in (either) of the forms (u_t + c*u_x)_t - c*(u_t + c*u_x)_x = - V(u), (u_t - c*u_x)_t + c*(u_t - c*u_x)_x = - V(u). Two families of characteristics, that interact with each other. Can no longer produce an exact solution (i.e.: cannot solve explicitly for u_t + u_x, or u_t - u_x, along their corresponding characteristics. However: 1. Examine how the solution advances in space time. 2. Notions of domain of dependence and domain of influence. 3. Light/sonic cone. No information can propagate faster than c. If solution vanishes outside an interval, it stays zero outside light cone later. % % -------------------------------------------------------------------- #066 D'Alembert's solution for the wave equation. Solve u_tt - c^2 u_xx = 0, with initial conditions u(x, 0) = U(x) u_t(x, 0) = V(x) using the general form of the solution u = F(x-c*t) + G(x+c*t), and show that u = (1/2)*[U(x-c*t)+U(x+c*t)] + (1/2*c)\int_{x-c*t}^{x+c*t} V(s) ds % % -------------------------------------------------------------------- #067 Wave equation on a finite interval 0 < x < L. Boundary conditions for a tied end string (= to pipe with closed end). """"""""""""""""""""""""" free end string (= to pipe with open end). Guitars and organs. % % ========================================================================== % ---------------------------------------------------- Lecture 20 Thu-Apr-26 % % -------------------------------------------------------------------- #068 Example: Solve the wave equation in 0 < x < L, for boundary conditions corresponding to a tied string of length L. Show that the solution is equivalent to a periodic (in space, of period = 2*L) and odd solution to the wave equation. --- extend solution by "reflecting" across the ends. Vice-versa: show that any periodic and odd solution of the wave equation provides a solution to this problem. NEW TOPIC: % ----------------------------------------------------------------------- % % % % Separation of variables and normal modes. % % % % ----------------------------------------------------------------------- % % % -------------------------------------------------------------------- #069 Normal modes. Equations of the form u_t = Lu Relationship with separation of variables: equation invariant under time shift allows separation u = exp(lambda*t) U(x) Example: write wave equation as u_t = v and v_t = u_xx. heat equation u_t = u_xx --- Note analogy with linear o.d.e. dY/dt = A*Y, A NxN matrix, solved by finding eigenvalues and eigenvectors of A. --- Hence look for solutions of the form u = e^{\lambda*t} v(x) Solve and find normal modes (eigenvalues and eigenfunctions). --- General solution: Superposition ... leads to Fourier Series, etc. Example: 1-D heat equation with zero temperature B.C. Associated with Sine Fourier Series. % % -------------------------------------------------------------------- #070 Why/how does this work? Review for matrices: NxN self-adjoint matrices have real eigenvectors and a complete base of orthonormal eigenvectors. NxN skew-adjoint matrices have pure imaginary eigenvectors and a complete base of orthonormal eigenvectors. Turns out that these results have infinite dimensional generalizations, as we will see below. Note: for general matrices the situation is more complicated. There are, usually, not enough eigenvectors to form a basis. "Generalized" eigenvectors are needed. That is, one must solve problems of the form (Jordan blocks) (A-lambda) u = 0, (A-lambda) v = u, (A-lambda) w = v, ... To obtain a basis. There is no good generalization of this to infinite dimensions. Note: Normal matrices are intermediate in complication between self-adjoint and generic. These matrices have a complete base of eigenvectors, and there is a good extension of the theory to infinite dimensions. A M matrix is normal iff M^ M = M M^, where M^ is the adjoint. % % -------------------------------------------------------------------- #071 To see how to extend the results in #070 to infinite dimensions, we need to first re-interpret matrices into a coordinate-free formulation: 1) Vector/linear spaces. 2) Interpretation of matrices as linear operators in a vector space. 3) Interpretation of self-adjoint for matrices in terms of the scalar product. % % ========================================================================== % ---------------------------------------------------- Lecture 21 Tue-May-01 % % -------------------------------------------------------------------- #072 Generalization of #071 4) Definition of scalar product and Hilbert space. 5) Self-adjoint and skew-adjoint operators in a Hilbert space. 6) Finite dimensional theory can be extended for these operators: All the eigenvalues are real (pure imaginary, resp.) and the eigenfunctions form a complete orthogonal set. [A subtle point swept under the carpet: In the examples below, the domain of definition of the various operators is not the whole space. It turns out that this is not a problem, but we will not get into the details.] % % -------------------------------------------------------------------- #073 Examples: --- Linear operator for the heat equation with various B.C. 1. In a ring: periodic. 2. Zero T at ends. 3. Zero flux at ends. 4. Robin conditions. In all these cases, get a self-adjoint operator, and various types of Fourier series to express the completeness of the eigenfunctions. % % -------------------------------------------------------------------- #074 Examples: --- Do some example for the wave equation, like tied ends. Use the formulation u_t = v_x and v_t = u_x. Skew-adjoint example. Free ends also gives skew-adjoint. Periodic also gives skew-adjoint. % % -------------------------------------------------------------------- #075 Will not cover [SKIP] Sturm-Liouville problems. Series and Transforms. % % ========================================================================== % ---------------------------------------------------- Lecture 22 Thu-May-03 % % -------------------------------------------------------------------- #076 Example: Wave equation, u_tt - u_xx = 0, with periodic B.C. Do directly in form above, and get double Fourier series. Use exponential notation, and show solution has form u = f(x-t) + g(x+t). % % -------------------------------------------------------------------- #070 % SKIP. Maybe assign in a problem set. Consider a string tied at the ends. Use a-dimensional variables. Then: u_tt - u_xx = 0 and u = 0 at x = 0 and x = pi Find normal modes (or separate variables), and find connection with characteristics: Normal modes as superpositions of a right and a left traveling wave. % % -------------------------------------------------------------------- #071 % SKIP. Maybe assign in a problem set. Wave equation. Using normal modes, show that: Boundary conditions for a tied string of length L lead to a solution of space period P = 2*L --- extend solution "reflecting" across ends. % % -------------------------------------------------------------------- #072 Example: 1) u_t = u_x, periodic of period 2*pi, 2) full line. Both have skew-adjoint operator [u_t = L*u] in appropriate L^2. Look for eigenfunctions: Case (1). Get complex, periodic, Fourier Series. Case (2). Get "nothing"!? There are no eigenvalues/eigenfunctions in the "traditional sense". What is going on? Review eigenvectors/eigenvalues for Matrices: Theorem for square matrices: (A-lambda) v = 0 has solution iff """""""" not invertible """""""" not injective """""""" not surjective. This fails in infinite dimension. Construct example of injective but not surjective: For periodic functions, map exp(i*n*x) to exp(i*2*n*x). Construct example of surjective but not injective: Use sin(n*x) base for L^2([0, pi]), and map sin((n+1)*x) to sin(n*x). Compute inverse of (L-lambda) in case (2) above. Note that inverse fails to exist if lambda is pure imaginary [precisely where "almost" eigenfunctions, bounded but not integrable, happen to exist]. General definition of spectrum, using failure of (A-lambda) to have an inverse. Continuum and Discrete spectrum [skip residual!!!] Show a decomposition in terms of "eigenfunctions" still exist for case (2) above, but with an integral over the spectrum replacing a sum over the eigenvalues: Get Fourier Transform by doing the limit of the periodic problem, as ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ the period \to infinity. That is: limit of Fourier Series. This idea in this approach can be extended to many other examples [see Coddington & Levinson] but we will not cover this here. % % ========================================================================== % ---------------------------------------------------- Lecture 23 Tue-May-08 % % -------------------------------------------------------------------- #073 Separation of variables. Product solutions. Skipped in the lectures, but problems assigned. READ THE SUPPLEMENTARY MATERIEL IN PROBLEM SET #5: Short notes on separation of variables. % % ------------------------------------------------------------------------- % ======================================================================= % % The material below is (partly) contained in the WEB page notes: % % Stability of Numerical Schemes for PDE's. % % Further stuff can be found in the introductions to the problem suites % % (these will be posted in course WEB page) % % --- GBNS Good and Bad Numerical Schemes. % % --- vNSA von Neumann Stability Analysis. % % --- AENS Long wave equation associated to a numerical scheme. % % % % This will take about 3-4 lectures, till the end of the semester. % % Lecture 23 Tue-May-08. % % Lecture 24 Thu-May-10. % % Lecture 25 Tue-May-15. % % Lecture 26 Thu-May-17. % % ======================================================================= % Finite difference schemes for PDE's. Follow notes in WEB page: --- Stability of Numerical Schemes for PDE's. 1 Naive Scheme for the Wave Equation. ............ lecture #16 USE the MatLab script in the 18311 toolkit: GBNS_lecture. 2 von Neumann stability analysis for PDE's. ...... lecture #17 3 Numerical Viscosity and Stabilized Scheme. ..... lecture #18 Reminder of details to include here: 1) von Neumann stability analysis Recall that solutions to time-evolution linear PDE's can be found by separation of the time variable --- analogy with ODE approach --- leading to an eigenvalue problem. Extend this idea to constant coefficients linear finite difference schemes -- key to the von-Neuman stability analysis. 2) Examine instabilities using associated equation: --- Explain behavior via forward and backward heat equations. --- Introduce stabilization by artificial viscosity (general idea). Relationship with the solutions of the heat equation by separation of variables exp(-k^2*t + i*k*x} 3) Define CONSISTENCY. von Neumann and consistency: Numerical and exact growth rates; comparison in the small k limit. 4) Define STABILITY. 5) Lax Theorem: for linear schemes, Consistency+Stability ==> Convergence. More details: Numerical and exact growth rates; comparison in the small k limit. Show that the general solution to a finite differences linear scheme can be written as a linear combination of the solutions G^j*e^(i*k*n) obtained by the von Neuman stability analysis. In other words, show that the matrix {w^{n*m}} has an inverse, where w = e^{i*2*pi/N} Associated equation. More examples of von Neuman stability analysis; associated equation, and stabilization by artificial viscosity: Lax-Friedrich sheme. % % ------------------------------------------------------------------------- % ========================================================================== % %% EOF