Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 1: 1.1, 1.2 1.1 ==== EXAMPLE. The advection equation (d/dt)u(x,t)+ (d/dx)u(x,t) = 0 SOLUTION: u(x,t) = f(x-t) where f is any differentiable function. EXAMPLE. The advection equation (d/dt)u(x,t)+ (d/dx)u(x,t) = 0 with initial condition u(x,0)=exp(-x^2). SOLUTION: u(x,t) = exp(-(x-t)^2) EXAMPLE. The advection equation (d/dt)u(x,t)+ kappa (d/dx)u(x,t) = - r u(x,t) MODEL. Assume wind blows in the positive x-axis direction at speed kappa m/s carrying along pollutant from a factory at the origin. Let u(x,t)=number of particles per meter at time t and assume (r)u(x,t) particles fall out of the air. DEF. The number of particles on 0<=x (alpha,beta) which writes the advection equation in the form (d/d alpha)U(alpha,beta)=0. Then U=constant that depends on beta, or u(x,t)=U(alpha,beta)=f(beta). The change of variables is alpha = ax + bt, beta = cx + dt The chain rule applied to these variables gives a new partial differential equation in u(alpha,beta): (a+b)u[alpha] + (c+d)u[beta] = 0 The symbols u[alpha] and u[beta] are the partials on u with respect to alpha and beta, resp. To eliminate one term, we take a+b=1 and c+d=0. Then u[alpha]=0, which implies u(alpha,beta)=function of beta alone. To further specify alpha and beta, we choose a=c=1, b=0, d=-1, then alpha=x, beta=x-t and finally u(x,t) = f(beta) = f(x-t) where f is a completely arbitrary function. 1.2 Vibrating string and Fourier series ==== The vibrating string u[t][t] = c^2 u[x][x] u(0,t)=0, u(L,t)=0, u(x,0)=f(x), u[t](x,0)=g(x) Special problem for f(x)=sin(Pi x/L), g(x)=0: u(x,t)=sin(Pi x/L) sin(Pi ct/L) Filmstrip of snapshots y=u(x,t) for t=0,0.25,0.5,0.75,1,1.25,1.5,1.75,2 See Figure 5 in the textbook Superposition u(x,t)= sum( b_n sin(n Pi x/L) sin(n Pi ct/L), n=1 .. infinity) Def. Fourier series of f(x)