1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20...
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
1) (15 marks) Consider the following PDHE Uz(0, t) = 0, u(5,t)=1, t>0 u(x, 0)- 20 exp(-2), 0<x<!5 (a) Solve using separation of variables. You may leave the eigenfunction expansion coef (b) Plot the solution at t-1,3,5 and 30, along with the initial condition and steady state ficients in inner product form. solution, using 15 terms in your truncated expansion. You may use mupad to evaluate the eigenfunction expansion coefficients from part (a) which you left in inner product form...
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0<2 (a) Find the steady state solution u,(x) of this problem. b) Write a new PDE, boundary conditions and initial conditions for U(x, t) - u(x, t)- Cox) (c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues...
2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and...
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral. 9. Use a suitable Fourier Transform...
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1 nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
Mark which statements below are true, using the following: Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx where FER is a constant, forcing term. Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two parts, taking into account that all change eventually dies out. That is there is a transient part...
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)...