Consider the heat equation below for a slab in 0x1 with a source term (Q =...

 

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Consider the heat equation below for a slab in 0≤x≤1 with a source term (Q = 4e²x), boundary conditions, and initial condition. Subscripts denote partial derivatives: u₁(x, t) = uzr(x, t) + 4e², 0≤x≤ 1, t> 0, boundary conditions ur(0, t) = 3, uz(1,t) = -1, initial condition u(,0) =sin Tx. (a) (4 points) For which value(s) of 3 does a steady state solution (equilibrium), ue(r), exist? (b) (5 points) Show that, for this value of 3, the quantity E(t) defined by = = [₁u(x, t) dr, E(t): is constant, dE/dt =0. Hence, E(x) = E(0). Hint: Integrate the PDE from x = 0 to 2 = 1 using the boundary conditions. (c) (5 points) You will note that the equilibrium ue(r), when it exists, is not fully deter- mined (it contains an unknown constant). Determine this unknown constant using the fact deduced above. Hint: As t→∞ the solution to the PDE will approach the equilibrium solution. So, E(oo) fue(x)dx, and you can calculate E(0) from the given initial condition. = Consider the heat equation below for a slab in 0≤x≤1 with a source term (Q = 4e²x), boundary conditions, and initial condition. Subscripts denote partial derivatives: u₁(x, t) = uzr(x, t) + 4e², 0≤x≤ 1, t> 0, boundary conditions ur(0, t) = 3, uz(1,t) = -1, initial condition u(,0) =sin Tx. (a) (4 points) For which value(s) of 3 does a steady state solution (equilibrium), ue(r), exist? (b) (5 points) Show that, for this value of 3, the quantity E(t) defined by = = [₁u(x, t) dr, E(t): is constant, dE/dt =0. Hence, E(x) = E(0). Hint: Integrate the PDE from x = 0 to 2 = 1 using the boundary conditions. (c) (5 points) You will note that the equilibrium ue(r), when it exists, is not fully deter- mined (it contains an unknown constant). Determine this unknown constant using the fact deduced above. Hint: As t→∞ the solution to the PDE will approach the equilibrium solution. So, E(oo) fue(x)dx, and you can calculate E(0) from the given initial condition. =

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The partial differential equation PDE given for the heat equation with a source term is utxt uxxxt 4e2x with boundary conditions ux0t beta quad ux1t 1 ... View the full answer