Chapter 6 Partial Differential Equations

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24 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS P 0 C 0 C 1 Example 6.2.16. Recall the Example 6.2.12 Solution Surface for Each Curve C 1 u t + cu x =0, u(x, 0) = F (x) with solution given by u(x, t) =F (x + ct). We note that the initial surface is t = 0 and the characteristics are determined by dx dτ = c, x(s, 0) = s, dt =1,t(s, 0)=0, ⇒ t(s, τ) =τ, x(s, τ) =cτ + s, ⇒ x − ct = s. dτ Thus we see that the solution is constant along characterisitics, i.e., (x, t) on a characteristic means that x − ct = s = constant and the solution is given by u(x, t) =F () and u is constant on a characteristic, i.e., a line x − ct = s for s ∈ R. A generalization of this problem is the case in which c = c(x, t). Then the characteristics are determined by and along this curve dx dτ = c(x, t), x(s, 0) = s, dt dτ =1,t(s, 0)=0, du dt (x(t),t)=u dx x dt + u dt t dt = u xc(x(t),t)+u t ≡ 0.
6.2. LINEAR AND QUASILINEAR EQUATIONS OF FIRST ORDER 25 Hence u is constant on a characteristic. Consider, for example, The characteristics are determined by which yields the parabolas u t +2tu x =0, u(x, 0) = e −x2 . dx dt =2t x = t 2 + k, k constant . The characteristic through a point (ξ,0) is x = t 2 + ξ. Since u is constant on this curve we have u(x, t) = exp(−ξ 2 )=e −(x−t2 ) 2 . Example 6.2.17. In the study of fluid flow an important physical characteristic is the formation of shock waves. The simplest example of the formation of shocks can be witnessed in the study of certain quasi-linear equations in R 2 called hyperbolic conservation laws. These are equations of the form subject to initial conditions The characteristics are determined by Along this curve u t + c(u)u x =0, x ∈ R, t>0 (6.2.16) u(x, 0) = ϕ(x), x ∈ R. (6.2.17) dx dt = c(u) du dt (x(t),t)=u x(x(t),t) dx dt + u t(x(t),t) dt dt = u xc(u)+u t ≡ 0. Hence u is constant on a characteristic. The characteristics are straight lines since d 2 x dt 2 = d dt ( ) dx = d dt dt c(u) =c′ (u) du dt =0,
Page 1 and 2: Chapter 6 Partial Differential Equa
Page 3 and 4: 6.1. INTRODUCTION 3 In this case we
Page 5 and 6: 6.1. INTRODUCTION 5 Solutions to La
Page 7 and 8: 6.1. INTRODUCTION 7 t u=0 u=1/2 u=1
Page 9 and 10: 6.2. LINEAR AND QUASILINEAR EQUATIO
Page 23: 6.2. LINEAR AND QUASILINEAR EQUATIO
Page 31 and 32: 6.3. CHARACTERISTICS AND HIGHER ORD
Page 53 and 54: Bibliography [1] R. Dautray and J.L

Chapter 6 Partial Differential Equations

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