(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u...
(1 point) Solve the heat problem with non-homogeneous boundary conditions ∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0 u(0,t)=0, u(3,t)=2, t>0,u(0,t)=0, u(3,t)=2, t>0, u(x,0)=23x, 0<x<3.u(x,0)=23x, 0<x<3. Recall that we find h(x)h(x), set v(x,t)=u(x,t)−h(x)v(x,t)=u(x,t)−h(x), solve a heat problem for v(x,t)v(x,t) and write u(x,t)=v(x,t)+h(x)u(x,t)=v(x,t)+h(x). Find h(x)h(x) h(x)=h(x)= The solution u(x,t)u(x,t) can be written as u(x,t)=h(x)+v(x,t),u(x,t)=h(x)+v(x,t), where v(x,t)=∑n=1∞aneλntϕn(x)v(x,t)=∑n=1∞aneλntϕn(x) v(x,t)=∑n=1∞v(x,t)=∑n=1∞ Finally, find limt→∞u(x,t)=limt→∞u(x,t)= Please show all work. (1 point) Solve the heat problem with non-homogeneous boundary conditions au ди (x, t) at (2, t), 0<x<3, t> 0 ar2 u(0,t) = 0, u(3, t) = 2, t>0, u(t,0)...
(1 point) Solve the heat problem with non-homogeneous boundary conditions du (x, 1) = ot (x,1), 0<x<2, t> 0 dx (0,t) = 0, (2, 1) = 2, t> 0, u(x,0) = 0<x<2. Recall that we find h(x), set u(x, t) = u(x, t)-h(x), solve a heat problem for u(x, t) and write u(x, t) = u(x, t) + h(x). Find h(x) h(x) = The solution u(x, t) can be written as u(x, t) = h(x) + u(x, t), where u(x,...
(1 point) Solve the heat problem with non-homogeneous boundary conditions v (2,t) = (2,t), 0<=<4, t>0 u(0,t) =0, u(4,t) = 2, t>0, ulz,0) = , 0 <I<4. Recall that we find h(2), set u(2,t) = u(2,t) – h(2), solve a heat problem for v(, t) and write u(2,t) = v(2,t) +h(2) Find h() (2) = The solution (I, t) can be written as uz, t) =h(2) + (,t), where (2,t) = »=Ecseh (a) v2,t) = Finally, find limu,t) = t-o
əz2(7,t), 0< < 4, t > 0 3 2,0<<< v(z,t) = { (1 point) Solve the heat problem with non-homogeneous boundary conditions ди au (2,t) at u(0,t) = 0, u(4, t) = 3, t > 0, u(2,0) 2,0<2<4. Recall that we find h(2), set v2,t) = u(2,t) – h(2), solve a heat problem for v2,t) and write uz,t) = v(x, t) +(2). Find h(1) h(x) = The solution u(x, t) can be written as u(x, t)=h(2) +v(2,t), where v(x, t)...
1. Consider the heat flow problem on the real line, where u(x,t), t > 0 is the temperature at point x at time t: ди 1 a2u t>O (*) at 2 ar2 u(x,0) = sin(7x) = > (a) What is the thermal diffusitivity constant ß? (b) Find the intervals of x where the temparature will increase at t = 0. (c) Sketch the graph of the temperature at t = 0. (d) On the same axes as in (c), sketch...
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...
Solve the heat flow problem: ди ди - (x, t) = 2 — (x, t), 0<x<1, t> 0, д дх2 и(0, t) = (1,1) = 0, t>0, и(x, 0) = 1 +3 cos(x) – 2 cos(3лх), 0<x<1.
Solve the heat flow problem: au t> 0, ди (x, t) = 2 (x, t), 0<x< 1, ot дх2 uz(0, t) = uz(1,t) = 0, t>0, u(x,0) = 1- x, 0 < x < 1.
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....
QUESTION 2 Consmder the problem ди 2k, 0<r< 1, t>O оt and the boundary conditions u(0,t)= 1, u (1,t) = 3, t > 0 (a) Find the equiltbrium solutiou ug (r) (b) Find the solution u (z.t) of the PDE and the boundary condition which also satisfies the mitial condition (,0)-1+++sin (3wx), 0<o< 1 [25]