Unformatted text preview: 1 such that
d2 u
− u = 0,
dx2 u(0) = 70, u(1) = 50. c. Solve the following special case of this boundaryvalue problem: Find u(x),
deﬁned for 0 ≤ x ≤ 1 such that
d2 u
+ x(1 − x) = 0,
dx2 u(0) = 0, u(1) = 0. d. (For students with access to Mathematica) Use Mathematica to graph the
solution to the following boundaryvalue problem: Find u(x), deﬁned for 0 ≤
x ≤ 1 such that
d2 u
+ (1 + x2 )u = 0,
dx2 u(0) = 50, u(1) = 100. You can do this by running the Mathematica program:
a = 0; b = 1; alpha = 50; beta = 100;
sol = NDSolve[ {u’’[x] + (1 + x∧2) u[x] == 0,
u[a] == alpha, u[b] == beta.}, u, {x,a,b}];
Plot[ Evaluate[ u[x] /. sol], {x,a,b}]
4.1.2.a. Show that the function
u0 (x, t) = √ 1 −x2 /4t
e
4πt is a solution to the heat equation (4.6) for t > 0 in the case where κ/(ρσ ) = 1.
¯
b. Use the chain rule with intermediate variables x = x − a, t = t to show that
¯
ua (x, t) = √ 1 −(x−a)2 /4t
e
4πt 84 is also a solution to the heat equation.
c. Show that ∞
−∞ u0 (x, t)dx = 1...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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