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....
Find the solution to the heat equation on the infinite
domain
∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1.
in terms of the error function.
Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
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]
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.
(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)...
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.
5. Find a solution u(x,t) of the following problem Ute = 2uz, 0< x < 2 u(0, t) u(2, t) = 0 u(x, 0) = 0, u(x, 0) = sin Tx - 2 sin 3ra .
5. Find a solution u(x,t) of the following problem Ute = 2uz, 0
Let u = u(x,y) and x = x(r,9), y = y(r,). ди ди a. Let x = r cos Q, y = r sin p. Find and a2u ar' 29 ar2 b. u = -x x = r sin 29,y = r tan’ 4, P (1,5). Find ou at the point P. де до
(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u (2,t), 0 < x < 5, t> 0 ar2 u(0,t) = 0, u5,t) = 3, t>0, u(x,0) = **, 0<x< 5. Recall that we find h(x), set v(x, t) = u(x, t) – h(x), solve a heat problem for v(x, t) and write u(x, t) = v(x, t) +h(x). Find h(c) h(x) = The solution u(x, t) can be written as u(x,t) =h(x) +...
= 0 over the domains 0<x<1 and t>0, where x is space and t is time at ax ди (1,1) = 0 ax Dirichlet and Neumann BCs are u(0, t)=80; Find the solution of the PDE that satisfies the given IC and BCs a. IC: u(x,0) 25sin (nx)