3. Let u(x, t) denote temperatures in a slab 0...

 

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3. Let u(x, t) denote temperatures in a slab 0<x< 1 that is initially at temperature zero throughout and whose faces are at temperatures u(0, 1) = 0 and u(1, 1) = F(t), where F(1) and F'(1) are continuous when 0 and where F(0) = 0. The unit of time is chosen so that the one-dimensional heat equation has the form u₁(x, t) = ux(x, 1). Write u(x, t) = U(x, t) + xF(t), and observe how it follows from the stated conditions on the faces of the slab that U(0, 1) = 0 and U(1, 1) = 0. Transform the remaining conditions on u(x, 1) into conditions on U(x, t), and then refer to the solution found in Problem 1 to show that 2 20 (-1)" u(x, t) = xF(1) + Σ TT n n=1 4. Show that when F(t) = At, where A is a constant, the expression for u(x, t) derived in Problem 3 becomes sin nπx x ['e = n² = ² (1-1)F' (7) dr. 0 2 u(x, 1) = A xt + Σ (−1)²¹- T n-1 - exp(-n²²1) n sin nπx 3. Let u(x, t) denote temperatures in a slab 0<x< 1 that is initially at temperature zero throughout and whose faces are at temperatures u(0, 1) = 0 and u(1, 1) = F(t), where F(1) and F'(1) are continuous when 0 and where F(0) = 0. The unit of time is chosen so that the one-dimensional heat equation has the form u₁(x, t) = ux(x, 1). Write u(x, t) = U(x, t) + xF(t), and observe how it follows from the stated conditions on the faces of the slab that U(0, 1) = 0 and U(1, 1) = 0. Transform the remaining conditions on u(x, 1) into conditions on U(x, t), and then refer to the solution found in Problem 1 to show that 2 20 (-1)" u(x, t) = xF(1) + Σ TT n n=1 4. Show that when F(t) = At, where A is a constant, the expression for u(x, t) derived in Problem 3 becomes sin nπx x ['e = n² = ² (1-1)F' (7) dr. 0 2 u(x, 1) = A xt + Σ (−1)²¹- T n-1 - exp(-n²²1) n sin nπx

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To transform the conditions on ux t into conditions on Ux t we substitute ux t Ux t xFt into the con... View the full answer