0, k > 0,IC: u(x, t = 0) = 9(x), 0 0.DE: ut =(a) Use the energy method to show that there is at most one solution for theinitial-boundary value problem.(b) Suppose u(x, t) = X (x)T(t) is a...

4. Consider the following initial value problem of the 1D heat equation with mixed boundary<br>condition<br>kuga, 0<x <I, t>0, k > 0,<br>IC: u(x, t = 0) = 9(x), 0<x < I,<br>ВС: и(0, t) — 0, и.(1,t) — 0, t>0.<br>DE: ut =<br>(a) Use the energy method to show that there is at most one solution for the<br>initial-boundary value problem.<br>(b) Suppose u(x, t) = X (x)T(t) is a seperable solution. Show that X and T satisfy<br>X

Extracted text: 4. Consider the following initial value problem of the 1D heat equation with mixed boundary condition kuga, 0 0, IC: u(x, t = 0) = 9(x), 0< i,="" вс:="" и(0,="" t)="" —="" 0,="" и.(1,t)="" —="" 0,="" t="">0. DE: ut = (a) Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b) Suppose u(x, t) = X (x)T(t) is a seperable solution. Show that X and T satisfy X" = XX, T' = \kT, for some A E R. (c) Find all the cigenvalues A, and the corresponding eigenfunction X, (r) for the problem X" = XX with boundary condition X (0) = 0, X'(1) = 0. (d) Solve the initial-boundary value problem. Express the solution as series of the eigenfunctions {X„(x)}.

Jun 05, 2022

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0, k > 0,IC: u(x, t = 0) = 9(x), 0 0.DE: ut =(a) Use the energy method to show that there is at most one solution for theinitial-boundary value problem.(b) Suppose u(x, t) = X (x)T(t) is a...

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