Question

Consider the following heat equation { u_t = ?^2 u_{xx}, 0 < x < L, t > 0 u_x(0,t) = 0, u_x(L,t) = 0, t ? 0 u(x,0) = f(x), 0 ? x ? L } (1) (6 pts) Use the method of separation of variables to reduce the above PDE to the following ODEs X'' + ? X = 0 X'(0) = 0, X'(L) = 0 and T' + ??^2 T = 0. (2) (6 pts) Knowing that the eigenvalues and eigenfunctions for X are ?_n = (n?/L)^2, X_n = cos(n? x/L), n = 0,1,2,... Use Question (1) to show that the solution of the above heat equation is of the form u(x,t) = a_0/2 + ?_{n=1}^? a_n e^{-(n? ? / L)^2 t} cos(n? x/L). (3) (8 pts) When ? = 1, L = ?, f(x) = x, use Question (2) to find the solution u(x,t) of the above heat equation with the given initial-boundary conditions.

          Consider the following heat equation
{ u_t = ?^2 u_{xx},  0 < x < L, t > 0
  u_x(0,t) = 0,  u_x(L,t) = 0,  t ? 0
  u(x,0) = f(x),  0 ? x ? L }

(1) (6 pts) Use the method of separation of variables to reduce the above PDE to the following ODEs
X'' + ? X = 0
X'(0) = 0,  X'(L) = 0
and
T' + ??^2 T = 0.

(2) (6 pts) Knowing that the eigenvalues and eigenfunctions for X are ?_n = (n?/L)^2, X_n = cos(n? x/L), n = 0,1,2,... Use Question (1) to show that the solution of the above heat equation is of the form
u(x,t) = a_0/2 + ?_{n=1}^? a_n e^{-(n? ? / L)^2 t} cos(n? x/L).

(3) (8 pts) When ? = 1, L = ?, f(x) = x, use Question (2) to find the solution u(x,t) of the above heat equation with the given initial-boundary conditions.

Consider the following heat equation
ut = ?^2 uxx, 0 < x < L, t > 0
ux(0,t) = 0, ux(L,t) = 0, t ? 0
u(x,0) = f(x), 0 ? x ? L

(1) (6 pts) Use the method of separation of variables to reduce the above PDE to the following ODEs
X” + ? X = 0
X'(0) = 0, X'(L) = 0
and
T' + ??^2 T = 0.

(2) (6 pts) Knowing that the eigenvalues and eigenfunctions for X are ?n = (n?/L)^2, Xn = cos(n? x/L), n = 0,1,2,... Use Question (1) to show that the solution of the above heat equation is of the form
u(x,t) = a0/2 + ?n=1^? an e^-(n? ? / L)^2 t cos(n? x/L).

(3) (8 pts) When ? = 1, L = ?, f(x) = x, use Question (2) to find the solution u(x,t) of the above heat equation with the given initial-boundary conditions.

Added by Joseph Y.

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