2. In lecture, we talked about the heat equation on a thin, laterally insulated rod. There...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and...
I having some trouble solving this boundary value problem. Thank you for your assistance. A thin wire coinciding with the x-axis on the interval [-L. L] is bent into the shape of a circle so that the ends x =-1 and x = L are joined. Under certain conditions, the temperature u(r, t) in the wire satisfies the boundary- value problem a au ot 11(-L. ) = u(Lt), t > 0 du ou .tso 11(x, 0) =fo), -L < x...
Consider the one-dimensional heat equation for nonconstant thermal properties debelo - (Kolon with the initial condition u(x, 0) = f(x). [Hint: Suppose it is known that if u(x, t) = ??(x ) h(t), then 1 dh 1 d do Ko(c) = -1 h dt c(x)p(x)o dx dx You may assume the eigenfuctions are known. Briefly discuss limt- u(x, t). Solve the initial value problem: (a) with boundary conditions u(0, t) = 0 and u(L, t) = 0 au *(b) with...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients 3. Consider the...
3. The last question is about physical interpretation of PDES. Assume throughout that u(r, t) describes the temperature in a thin rod, with r [0, L] andt2 0. Note: when I ask for physical interpretation, I mean things like "u(0,t) = 0 means that fixing one of the ends at constant temperature" (a) Consider the IBVP we are O (3 points) at Or2 u(0, t) Ou (L, t) u(x,0) =0 Give a physical interpretation of the boundary conditions and of...
11. Consider a thin, infinitely long rectangular plate that is free of heat sources, as shown below. For a thin plate, is negligible, and the temperature is a function of x and y only. The solution for this problem is best obtained by considering scaled temperature (ie. 1-T - To, where To is the absolute temperature at T-0) variables, so that the two edges of the plate have "zero-zero" boundary conditions and the bottom of the plate is maintained at...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where ?(?,?) represents the temperature. 9??? = ?? ; 0 < ? < 6; ? > 0; B. C. : ?? (0,?) = 0; ?? (6,?) = 0; ? > 0; I. C. : ?(?, 0) = 12 + 5??? ( ? 6 ?) − 4???(2??); 0 < ? < 6 (a) When ? = 0, what would be the temperature at ? = 3? (Use...