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Problem 33 Solve the boundary value 1D heat problem with the given data. In each case,...
Section 1.3 3. a. Solve the following initial boundary value problem for the heat equation 0x<L t0 at u(r, 0) f() u(0, t)u(L, t) 0, t>0, 9Tr when f(r)6 sin L b. Solve the following initial boundary value problem for the diffusion equation au D 0 L t0 at u(r, 0) f() (0, t) (L, t) 0, t 0, x < L/2 0. when f(r) r > L/2. 1 Section 1.3 3. a. Solve the following initial boundary value problem...
Please explain it thoroughly. 1D string and heat conductor Problem 1.1. (4 pts) Consider the 1D vibrating string equation ch.. (t,x) = hn(t,x) + fh(t,z), x E [0, L], f > 0 with the boundary condition h(t,0-0, hz(t, L)=0. Write the most general solution and discuss the qualitative behaviour of the solution (especially its depen- dence on t), can you give a physical interpretation to the f-term? Problem 1.2. (2 pts) Continuing from the above exercise. Now add the initial...
PDE’s 1. (a) Reduce the boundary value problem for 3D heat conduction with spherical symmetry u(z, y, 2, t-u(r, t f(r) is given] to the boundary value problem for 1D heat conduction in a rod with insulated latteral surface (b) Derive し2 u(r, t) = Σ Bn sin nm-e-n2 ' , where Bn=2 ,rf(r) sin nTI, dr. τ= (c) Suppose a ball (radius L-0.1) of molten aluminum (at its melting point) is dropped into freezing water Estimate how long it...
In Exercises 11-15, solve the nonhomogeneous wave initial-boundary-value problem. In each case, start by letting u(x,t) = T.(t) sin nz and proceed from there. n=1 11. u = Una + sin , u(,0) = sin 3.0, U (2,0) = sin 52, u(0,t) = u(Tt, t) = 0.
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
6. Solve the following boundary value problem: 1 U = 34xx, 0 < x < 1,t> 0; u(0,t) = u(1,t) = 0; u(x,0) = 7 sin nx - sin 31x
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0<x<L (a) Give a short physical interpretation of this problem. (b) Given the following initial condition, 2 *2 2 solve the initial boundary value problem for u(x,t. 3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0
PDE Problem: homogenous diffusion equation with non-homogenous boundary conditions 27. Solve the nonhomogeneous initial boundary value problem | Ut = kuzz, 0 < x < 1, t > 0, u(0, t) = T1, u(1,t) = T2, t> 0, | u(x,0) = 4(x), 0 < x < 1. for the following data: (c) T1 = 100, T2 = 50, 4(x) = 1 = , k = 1. 33x, 33(1 – 2), 0 < x <a/2, /2 < x < TT, [u(x,...
Solve the initial-boundary value problem for the following equation U = N Ux with U(x, 0) = sin (x) +N ,U(0, t) = 0, and U, (N, t) = 0 Q4| (5 Marks) my question please answer Solve the initial-boundary value problem for the following equation U = N Ux with U(x, 0) = sin (x) +N ,U(0, t) = 0, and U, (N, t) = 0 Q4| (5 Marks) Solve the initial-boundary value problem for the following equation Uų...
(4 points) This problem is concerned with solving an initial boundary value problem for the heat equation: u,(x, t)- uxx(x,), 0