Chapter 6 Partial Differential Equations

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4 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS Linear partial differential equation’s are often classified as being of elliptic, hyperbolic, or parabolic type. What follows is a brief and heuristic discussion of some of the features that characterize these classifications of partial differential equations. Most of what is presented in this introduction will be made precise in subsequent chapters. Elliptic <strong>Equations</strong> An example of an elliptic partial differential equation is Laplace’s equation, div grad u =0 If u = u(x 1 , ··· ,x n ) and (x 1 , ··· ,x n ) are rectangular Cartesian coordinates, and w(x) = (w 1 , ··· ,w n ), grad u = ∇u =(u x1 , ··· ,u xn ) div ⃗w = ∂w + ···+ ∂w ∂x 1 ∂x n and so div grad u = ∇·∇u = ∇ 2 u =∆u = u x1 x 1 + ···+ u xnx n =0. The Laplacian of u, ∆u, provides a comparison of values of u at a point with values at neighboring points. To illustrate this idea consider the simplest case that u = u(x) and assume u xx (x) > 0. Let ũ denote the tangent line approximation to u. For h>0 and sufficiently small, u(x + h) > ũ(x + h) =u(x)+u ′ (x)h u(x − h) > ũ(x − h) =u(x) − u ′ (x)h and so u(x + h)+u(x − h) >u(x). 2 Roughly stated, u(x) is smaller than its average value at nearby points if u xx (x) > 0. In higher dimensions, say n = 2, the analogous statement is that if ∆u(x, y) > 0, then the average value of u at neighboring points, say on a circle about (x, y), is greater than u(x, y). If ∆u(x, y) < 0, then u(x, y) is greater than the average value of u on a circle about (x, y). If ∆u(x, y) = 0 then u(x, y) is equal to its average value on a circle about (x, y). (We will subsequently prove a theorem that makes these ideas rigorous.) More generally, if ∆u(x) =0,x∈ Ω ⊂ R n , then u is equal to its average value at neighboring points everywhere in Ω. In a certain sense, this says that if u satisfies Laplace’s equation then u represents a state of equilibrium.
6.1. INTRODUCTION 5 Solutions to Laplace’s equation ∆u(x) = 0 are said to be harmonic. Suppose f(z) = f(x + iy) =u(x, y)+iv(x, y) is analytic. The Cauchy-Riemann equations state that ∂u ∂x = ∂v ∂y , ∂u ∂y = −∂v ∂x and so u xx + u yy =0, v xx + v yy =0. Thus the real and imaginary parts of an analytic function are harmonic. There is a converse statement of this result known as Weyl’s Theorem that depends on the notion of a weak solution. Roughly stated, ∆u(x) = 0 in a weak sense if for all smooth functions φ with compact support in Ω, ∫ u(x)(∆φ)(x) dx =0. We will not prove this next result. Ω Weyl’s Theorem If u is a weak solution of ∆u(x) = 0 on Ω, then u ∈ C ∞ (Ω) and u satisfies Laplace’s equation in a classical sense. The so-called Cauchy problem for Laplace’s equation, u xx + u yy =0, |x| < ∞,y >0 u(x, 0) = f(x), u y (x, 0) = g(x), is not well-posed. Indeed, consider the specific problem u xx + u yy =0, |x| < ∞,y >0 cos nx u(x, 0) = n u y (x, 0)=0 with the solution u n (x, y) = 1 cosh(ny) cos(nx). n For n sufficiently large, the data of this problem can be made uniformly small. However, lim u n(x, y) =∞. n→∞ In other words, small changes in the data do not correspond to small changes in a solution.
Page 1 and 2: Chapter 6 Partial Differential Equa
Page 3: 6.1. INTRODUCTION 3 In this case we
Page 7 and 8: 6.1. INTRODUCTION 7 t u=0 u=1/2 u=1
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Page 31 and 32: 6.3. CHARACTERISTICS AND HIGHER ORD
Page 53 and 54: Bibliography [1] R. Dautray and J.L

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