Determine the time-independent solution to the following BVIVP consisting of the PDE Ot for 0
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a) 14 points Consider the following equation : PDE: u+ 0 ,0
Find solution to the IBVP PDE BCs Ic u(0, t)-0, 0<oo l u(1, t) 0, 0<t< oo u(z,0)=x-x2ババ1
4. (XX pts.) Find solution to the IBVP PDE BCs IC a(0, t) = 0, a(1,t)=0. () < t < oo () < t
Assignment 0220 Marks) Solve the following IVBP: PDE : Uxx = (1/25) utt ICs: u (x,0) = x2 (nt - x), ut (x,0) = sin(x) BCs: u(0,t) = 0, u(nt,t) = 0 for 0<x<, t> 0. for 0<x<T. for t>0.
Problem # 1 [15 Points] Consider the following PDE which describes a typical heat-flow problem PDE: ut = ↵2uxx, 0 < x < 1, 0 < t < 1 BCs: ux(0, t)=0 ux(1, t)=0 0 < t < 1 IC: u(x, 0) = sin(⇡x), 0 x 1 (a) What is your physical interpretation of the above problem? (b) Can you draw rough sketches of the solution for various values of time? (c) What about the steady-state temperature?
Can you help me with this question please? (2) [15pts] Write the solution of the PDE for 0 <<1 and t>0 u(0,t) 0 u(1,t) = 0 u(x,0)-0 Make sure you simplify as much as possible by using the fact that the source term f depends only on z and not on t. What is the solution at long time lim,-+oo u(x, t) if f()T) [Hint: You can obtain this by solving a single (trivial) ODE] (2) [15pts] Write the solution...
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...
Please detail Please detail PDE Utt = Uzx + 2a sin(at) sin(1x) 0 < x <1 0<t< oo BCS S u(0,t) = 0 | u(1,t) = 0 0<t< oo ICs u(x,0) = 0 | u4(,0) = sin(nx) 0 < x <1 u (0,t) = f (t) u (L,t) = g(t) S Use sine transform Uz (0,t) = f(t) uz (L,t) = g(t)) Use cosine transform 2 L S [u (x,t)] = Sn (t) = 1 | u(x, t) sin (ntx/L)...
Solve the BVP: от от OT PDE: BC: T(0.t) = (5,0) = 0 (5.1) = 0 0SX 55 Or? BC: TO and T(x,0) = 1-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)