FW: Math_Partial Differential Equations

 

 

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Posted on: Thursday, February 04, 2010 7:02 PM
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Subject: Math_Partial Differential Equations

 

CHAPTER 1 The Physical Origins of Partial Differential Equations 1. Mathematical Models 1 Exercise 1. The verification that u = √4πkt e−x /4kt satisfies the heat equation ut = kuxx is straightforward differentiation. For larger k, the profiles flatten out much faster. 2 Exercise 2. The problem is straightforward differentiation. Taking the derivatives 1 is easier if we write the function as u = 2 ln(x2 + y 2 ). Exercise 3. Integrating uxx = 0 with respect to x gives ux = φ(t) where φ is an arbitrary function. Integrating again gives u = φ(t)x+ψ(t). But u(0, t) = ψ(t) = t2 and u(1, t) = φ(t) · 1 + t

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