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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...
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,...
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) ,...
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) =...
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...
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,...
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
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...
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...
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) =...
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
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.
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
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