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  3. 3 suppose ur t denotes the density of cars on...

Question: 3 suppose ur t denotes the density of cars on...

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3. Suppose u(r, t) denotes the density of cars on a certain stretch of highway (in units of cars per mile (a) Explain why u(r,t) obeys the fundamental conservation law (b) If we further assume that f0 in our conservation law equation, what does that mean in this (c) Assume f(x,t)0. One proposal is to model the lux of the cars with the state equation (d) Assume f(rt)0. For the flux term proposed in (c), what is the resultant PDE which models scenario? ф (M-u)2 where M is the maximum capacity of the highway. Discuss if this is reasonable assumption for φ. the traffic density? Discuss (a)-(d)

Introduction to PDEs 2 How PDEs Are Born: Conservation Laws, Fluids, and Waves 1.2 How PDEs Are Born: Conservation Laws, integrals are equal for an arbitrary subinterval la, b) of 0 <t, the integrands must be equal, producing the Fundamental Conservation Laur Fluids, and Waves DE(z,t) z(z,t) = f(z, t). Conservation Laws and Fluids Consider fluid flow in a one dimensional domain (e.g., heat flow in a very thin rod or Let t >0 denote time and 0 <エ< 1 denote the spatial variable. Define the following However, the Fundamental Conservation Law alone is not enough to derive a partial differential equation modeling heat fow or chemical diffusion because there are still two unknowns-u and p (the source term f is typically thought of as given). Thus, we appeal to further physical assumptions on the fux term o called constitutive equations n of dye in a very thin tube of fluid) of length as shown in Figure 1.4 quantities: u(r,t)the density of the quantity of interest from fluid dynamics piz,t):- the flux of the quantity at the point a and time t (the amount of the f(r,t): rate ofinternal generation of the quantity at the point z and time t The following are some physically realistic constitutive equations that lead to wel studied PDEs (eg., heat energy or dye concentration) at the point r and time t flowing to the right per unit time per unit area through a cross section at z) amount per unit volume per unit time) pcu, f 0. Then (1.3) becomes called the transport equation because it models the transport of a (nondiffusing) substance by a flowing fluid. Acros-sectional area of a slice of the domain (assumed to be the same for all r) . p -kuz. f0. Then (1.3) becoes kur0, or equivalently kuza called the heat equation or diffusion equation since this constitutive equation is consistent with Fouriers Law of Heat Conduction and Ficks Law of Diffusion . ф cu-ku,, f . 0. Then (1.3) becomes Figure 1.4: Modeling fluid flow in one space dimension. Consider an arbitrary subinterval (a, bl of the domain 0 < cross-sectional area A. The conservation law here is t with (constant) called the transport-diffusion equation because it models the combined dynamics of a diffusing substance in a flowing fluid. time rate of change rate of internal of total quantity in a slicerate of inflow rate of out low + ·p-u2/2, f generation. 0, Then (1.3) becomes Translating this into symbols i the inviscid Burgers equation, which arises in fuid dynamics. p2/2-ku f 0. Then (1.3) becomes called Applying the Fundamental Theorem of Calculus to the difference on the right-hand We can rewrite this as the integral equation called the viscous Burgers equation and arises in more advanced fluid dynamics. Also called Heat flows down the time rate with negligible viscous (frictional) forces are called inviscid

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