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
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variables without making any preliminary trans- formation. Does your solution agree with the solution you would obtain if transformation u(x, t)= e(caret)(-t) w(x, t) were made in advance?
Solve the PDE using laplace: (dw/dx)+(x*(dw/x)) =xu(t-1) w(x,0)=0 if x>=0 and w(0,t)=0 if t<=0
1-4 Determine whether each PDE can be separated. Then separate it into two ODEs if it is possible to separate. x u xt 1-4 Determine whether each PDE can be separated. Then separate it into two ODEs if it is possible to separate. x u xt
What type of PDE is this? Solve PDE using separation of variables (show all the work and logic) 05 x u(x,0) 4sin(37r), u,(x,0) 2sin(57) 0sx 1,t 2 0
PDE: Ut = Uxx, -00 < x < 0, t> 0 IC: u(x,0) = 38(x) + 28(x – 6) where is the Dirac delta function (impulse). u(x, t) =
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
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
1-4 Determine whether each PDE can be separated. Then separate it into two ODEs if it is possible to separate. x u xt
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...