Solve the heat equation u t = k 2u x2 . [Hints: It is known that if u(x, t) = (x) G(t)
Chapter 2, Problem 2.4.7(choose chapter or problem)
Solve the heat equation u t = k 2u x2 . [Hints: It is known that if u(x, t) = (x) G(t), then 1 kG dG dt = 1 d2 dx2 . Appropriately assume > 0. Assume the eigenfunctions n(x) satisfy the following integral condition (orthogonality): L 0 n(x)m(x)dx = 0 n = m L/2 n = m subject to the following conditions: (a) u(0, t) = 0, u(L, t) = 0, u(x, 0) = f(x) (b) u(0, t) = 0, u x (L, t) = 0, u(x, 0) = f(x) (c) u x(0, t) = 0, u(L, t) = 0, u(x, 0) = f(x) (d) u x (0, t) = 0, u x(L, t) = 0, u(x, 0) = f(x) and modify orthogonality condition [using Table 2.4.1.]
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