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Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept at 0 for all t, and its temperature distribution at time t0 is given by T(a,0)a. Find the temperature function T(,t) for the rod
3. 3 points A metal rod has a length of 10cm, and has its left end...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t)...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Uti 0<x< 6; t> 0; B.C.: ux(0,t) = 0; ux(6,t) = 0; t> 0; 1. C.: u(x, 0) = 12 + scos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t)...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0<x< 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t> 0; I. C.: u(x,0) = 12 + 5cos (x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t)...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Ut; 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz(6,t) = 0; t> 0; 1. C.: 4(x,0) = 12 + Scos (6x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t)...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t) represents the temperature. 9uxx = upi 0<x< 6; t> 0; B.C.: uz(0,t) = 0; ux, t) = 0; t> 0; 1.C.:u(x,0) = 12 + 5cos 6 x) – 4cos(26x); 0<x<6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or mixed...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t)...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t) represents the temperature. 9uxx = ut; 0<x< 6; t> 0; B.C.: Ux(0,t) = 0; Ux(6,t) = 0; t> 0; 1.C.: u(x,0) = 12 + Scos (x) – 4cos(21x); 0 < x < 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet,...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = uti 0 < x < 6; t>0; B.C.: 4x(0,t) = 0; uz (6,t) = 0; t> 0; 1. C.: u(x,0) = 12 + 5cos (6x) – 4cos(27x); 0<x<6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0 < x < 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t>0; I. C.: u(x,0) = 12 + 5cos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
3. 3 points A metal rod has a length of 10cm, and has its left end at zn0 and right end at z = 5. The temperature T(z,1) of the rod satisfies the heat equation T,r(z,t)-m(r,t), its ends are kept at 0 for all t, and its temperature distribution at time t0 is given by T(a,0)a. Find the temperature function T(,t) for the rod
3. 3 points A metal rod has a length of 10cm, and has its left end...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Uti 0<x< 6; t> 0; B.C.: ux(0,t) = 0; ux(6,t) = 0; t> 0; 1. C.: u(x, 0) = 12 + scos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0<x< 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t> 0; I. C.: u(x,0) = 12 + 5cos (x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = Ut; 0<x< 6; t> 0; B.C.: 4x(0,t) = 0; uz(6,t) = 0; t> 0; 1. C.: 4(x,0) = 12 + Scos (6x) – 4cos(27x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t) represents the temperature. 9uxx = upi 0<x< 6; t> 0; B.C.: uz(0,t) = 0; ux, t) = 0; t> 0; 1.C.:u(x,0) = 12 + 5cos 6 x) – 4cos(26x); 0<x<6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann, or mixed...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,t) represents the temperature. 9uxx = ut; 0<x< 6; t> 0; B.C.: Ux(0,t) = 0; Ux(6,t) = 0; t> 0; 1.C.: u(x,0) = 12 + Scos (x) – 4cos(21x); 0 < x < 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is Dirichlet,...
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = uti 0 < x < 6; t>0; B.C.: 4x(0,t) = 0; uz (6,t) = 0; t> 0; 1. C.: u(x,0) = 12 + 5cos (6x) – 4cos(27x); 0<x<6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (3 marks) (b) Determine whether the boundary conditions in this case is...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0 < x < 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t>0; I. C.: u(x,0) = 12 + 5cos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
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