1. Solve the following transient heat equation by separation of variable method. aT At x = 0, BC み_Qo Atx=L, T =To IC: At t = 0, T= 1. Solve the following transient heat equation by separat...
1. Solve the following transient heat equation by separation of variable method. ат ах Atx=L, T =To IC: At t = 0, T=T, 1. Solve the following transient heat equation by separation of variable method. ат ах Atx=L, T =To IC: At t = 0, T=T,
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12. Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
ONLY ANSWER 5 and 9. Rating will be provided Exercises for Section 6.2 In Exercises 1-12 use the separation of variables method to solve the heat equation (a, t)auz(t<<l,t>0, subject to the following boundary conditions and the following initial conditions: a = V2, l = 2, u(0,t) = u(2,t)=0, and 5. 20, 0r< 1 0, a(x, 0) = rS 2. 1 1 = π, u(z, 0) = π-z, u(0, t) = uz(mt) = 0. 9. Exercises for Section 6.2 In...
3. Determine the discretization of the heat equation driven at the boundary, PDE IC (x,0) BC t E R+ u(0, r) =g(x),ur(1,1) = 0, 3. Determine the discretization of the heat equation driven at the boundary, PDE IC (x,0) BC t E R+ u(0, r) =g(x),ur(1,1) = 0,
1. Solve the vibrating string problem PDE BC BC IC IC utt T.T Uz(0,t) = 0 u(1,t0 u(x,0cos(3T) 14(2.0) = x
Solve the heat equation by the method of separation of variables 3π u(x,0)--2cos( x)
2. For the 1-D heat equation solve uha, t) with Cs and ICs wing seperating BC (0,0) = 0, Lt) = 0 ICs (2,0) = cos 21 c20²u a2x' au 2. For the 1-D heat equation solve u(x, t) with BCs and ICs using separating at variables. Please show the details. BCs: & u(0,t) = 0, u(L, t) = 0 ICs: u(x,0) = cos Sex 2L
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) = 3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
Explain how does w(x) been solve Decomposing inhomogeneous PDEs to facilitate the use of separation of variables Inhomogeneities may arise in the initial (ICs) or boundary (BCs) conditions, or in the PDE itself. A simple example is the falling of an elastic wire under gravity: ə?u ,02u at2 = Car2g If the ICs are: u(x,0) = f(x) and (x,0) = 0, and the BCs are: u(0,t) = 0 and u(L,t) = h(t), then there are three inhomogeneities in this equation:...
3. Using separation of variables to solve the heat equation, u -kuxx on the interval 0x<1 with boundary conditions u(0 and ur(1, t)-0, yields the general solution, u(x, t) =A0 + Σ Ane-k,t cos(nm) (with A, = ㎡π2) 0<x<l/2 0〈x〈1,2 u(x,0)=f(x)-.., , . . .) when u(x,0) = f(x)- Determine the coefficients An (n - 0, 1,2,