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(a) The temperature distribution u(x, t) of the one- dimensional silver rod is governed by the...
(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...
For : U(x,0) = Sin(ax) a= 2.6 using the Explicit Forward Euler and Crank-Nicholson methods. Example 92. One-Dimensional...
For : U(x,0) = Sin(ax) a=
2.6
using the Explicit Forward Euler and Crank-Nicholson
methods.
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2) We were unable to transcribe this image
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2)
Question 1: The separated solutions of the o fom u(x.t) -X(x)T(t), with the following solutions: ne-dimensional he...
Question 1: The separated solutions of the o fom u(x.t) -X(x)T(t), with the following solutions: ne-dimensional heat equation dtt lu solutions of are - X(x)-Ax +B and T(t) E X(x) = A cos kx + B sin kx and T(t)=Ee-Det The boundary conditions for a metal rod insulated from both sides arex aum = 0 when x =0, and dx (e) Using the boundary conditions for u(x.t) wrie the boundary conditions for XCx), explain for full marks. (b) Find the...
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...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 < x < L. Assume that the temperature at the end x=0 is held at 0°C, while the end x=L is thermally insulated. Heat is lost from the lateral surface of the bar into a surrounding medium. The temperature u(x, t) satisfies the following partial differential equation and boundary conditions aluxx – Bu = Ut, 0<x<l, t> 0 u(0,t) = 0, uz (L, t)...
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,...
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...
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to th...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find 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...
For : U(x,0) = Sin(ax) a=
2.6
using the Explicit Forward Euler and Crank-Nicholson
methods.
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2) We were unable to transcribe this image
Example 92. One-Dimensional Parabolic PDE: Heat Flow Equation. Consider the parabolic PDE d-u(x, t) du(x, t) 0t with the initial condition and the boundary conditions (E9.2.2)
Question 1: The separated solutions of the o fom u(x.t) -X(x)T(t), with the following solutions: ne-dimensional heat equation dtt lu solutions of are - X(x)-Ax +B and T(t) E X(x) = A cos kx + B sin kx and T(t)=Ee-Det The boundary conditions for a metal rod insulated from both sides arex aum = 0 when x =0, and dx (e) Using the boundary conditions for u(x.t) wrie the boundary conditions for XCx), explain for full marks. (b) Find the...
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...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 < x < L. Assume that the temperature at the end x=0 is held at 0°C, while the end x=L is thermally insulated. Heat is lost from the lateral surface of the bar into a surrounding medium. The temperature u(x, t) satisfies the following partial differential equation and boundary conditions aluxx – Bu = Ut, 0<x<l, t> 0 u(0,t) = 0, uz (L, 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) 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,...
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...
Fourier Wave Equation
Question 2 The function u(x,t) is governed by the wave equation 82u 182u 8x2 c2 8t2 Subject to the following conditions having c2 3 i. At x 0 and x-1, u 0 for all t 2 0. i. Whent 0, for sx s 1 Use the method of separation of variables to establish that the solution for u(x, t). From the solution established, given a condition t0, usinx(1 + cosx) for 0 s x s1. Find the...
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