Would like the full steps to get to the answer. Thank you.
Would like the full steps to get to the answer. Thank you. 6. Find a solution u(x, t) of the following problem 9u 4 0 U...
Problem 2. Find the type, transform to normal form, and find the solution u(x,t) of the ID wave equation, Utt = Uxx, with the initial conditions u(x,0) = 2sin 2x and ut(x,0) = 0 and the boundary conditions u(0,t) = u(nt,t) = 0.
5. Find a solution u(x, t) of the following problem utt 0 u(0, t) — и(2, t) — 0 2 sin 3T и(а, 0) — 0, и (х, 0) — sin Tz _
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...
4. Consider the semi-infinite string problem given by Utt = cʻuza, 0<x< 0,> 0 u(x,0) = f(x), 0<x< ~ ut(2,0) = g(2), 0 < x < 0 u(0,t) = 0, t> 0 Suppose that c=1, f(0) = (x - 1) - h(2 – 3) and g(C) = 0. (a) Write out the appropriate semi-infinite d'Alembert's solution for this problem and simplify. (b) Plot the solution surface and enough time snapshots to demostrate the dynam- ics of the solution.
If you were to solve the variant of wave equation utt=uxx+u for 0<x<6 and t>0 with u(0,t)=u(2 ,t)=0, u(x,0)=2x, ut(x,0)=0 using separation of variables, what would be the correct form of Xn (x)? Xn (x)=cosh( nπ 4 Xn (x)=sin( nπ 2 Xn (x)=sin( n2 π2 4 Xn (x)=cos nπ 2 None of these
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
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...
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)...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
3. (20 pts). Find the solution to the vibrating-string problem: utt u(0,t) u(L,t) u2,0) 2,0) = 0 = 0 2 sin(27/L) + sin(31/L) sin(72/L) 0<<L, 0<t< 0<t< oo 0<t< 0<r<L 0<r<L