5. Solve IBVP 11(0,1)-α, u(L,t)-β u(x,0)- f(x) 120 0Sx SL b) u-100, β-100, f(x)-50x( l-x), L-1,...
Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R. Problem 2.7.26. Solve the parabolic problem ubject to the nonhomogeneous boundary conditions u(t,0)-1 and u(t,1)or 0 and the initial condition u(0,x)(x for xE(0,1) for some given function f:(0,1) R.
2. Use eigenfunction expansion to solve the following IBVP: u,(x, t) ="-(x,t) + (t-1)sin(m), 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
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
g) Consider the problem Ou(x, t) = Oxxu(x, t), u(x,0) = Q(x), 0,u(0,1) = 0,1(L,t) = 0, (x, t) (0, L) x (0,00), T ( [0, LG, te [0,00). with a given function 0. Show that the energy L 1 ENE() = 1 u? (x, t)da decays in time.
please and thank you 1. Solve V2u= 0 over 0Sx<L and 0sysH subject to (1) 10% u(0, y)-0, u(L, y)-0, u(x,0)-0 and u(x, H)x (2) 15% u(0,y)-0, u(L,y)=y, u(x,0)=0 and u(x, H)-x. [Hint: superposition] 1. Solve V2u= 0 over 0Sx
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0<x <c,0 11 (c, 1) = 0, u(x, 0) = f(x), where u is continuous for0sxc,0 and where n is a positive integer. Answer: u(x, 1) Σ A,Jn(gjx) exp (-α,1), where a", and A, are the constants j-1 11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0
Solve the IBVP wave equation. d^2/dt^2=16d^2/dx^2 0<x<pi u(x,0)=sinx du(x,0)/dt=0 u(0,t)=u(pi,t) =0 t>0
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t) + 2t sin (2na:) , 0 < x < 1, 0, u(1,t)=0, t > 0, sin(2π.r)-5 sin (4π.r) , 0 < x < 1. t > 0, = = = 4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t)...
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...