Solve the heat equation Ut = Uxx + Uyy on a square 0 <= x <= 2, 0<= y<= 2 with the following boundary and initial conditions 2. Solve the heat equation boundary conditions uvw on a square O S r s 2, 0 S vS 2 with the (note the mix of u and tu) and with initial condition 0 otherwise Present your answer as a double trigonometric sum. 2. Solve the heat equation boundary conditions uvw on a...
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,
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...
[points=4] Q4. Solve the heat equation subject to the given conditions: д?u ди О<x<п, to дх? at' ди ди - (0,t) = 0, - (п,t) = 0, t>0 дх дх u(x,0) = п-х Paragraph В І := =
2. For the 1-D heat equation solve uha, t) with Cs and ICs wing seperating BC (0,0) = 0, Lt) = 0 ICs (2,0) = cos 21 c20²u a2x' au 2. For the 1-D heat equation solve u(x, t) with BCs and ICs using separating at variables. Please show the details. BCs: & u(0,t) = 0, u(L, t) = 0 ICs: u(x,0) = cos Sex 2L
Question 8 [1.5 mark] Solve the heat equation (0,0) = u(2,0) = 0, (u(x,0) - 2 sin (*) - sin(mr) + 4 sin (22) for u = u(x, t): 0,2 x 10,00) +R, using the method of separation of variables.
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0 1. Consider the Partial Differential Equation ot u(0,t) =...
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1, t> 0 u(x, 0, 1) = 0, uy(x,1,1) = 0, with boundary conditions u(O, y,t) = 0, u(r, y, t) = 0, and initial condition u(x, y,0) = (y – į v?) sin(2x).
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT EQUATION 1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...