Partial differential equation-heat equation,
The temperature distribution Θ(x, t) along an insulated metal rod of
length L is described by the differential equation.
The rod is held at a fixed temperature of 0◦ C atone end and is insulated at the other end, which gives rise to the boundaryconditions Θ(0, t) = 0 and Θx(L, t) = 0, for t > 0.
Show that function Xn(x) satisfies the boundary conditions that you found. Show that Xn(x) satisfies differential equation (1) for some constant µ (which you should specify).
Use your answers to write down a family of product solutions
Θn(x, t) = Xn(x) Tn(t) that satisfy the first two boundary conditions.
Homework Answers
This is a question about Partial differential equation - Heat equation. Please help solving part (a)...
This is a question about
Partial differential equation - Heat equation. Please help solving
part (a) and show clear explanations. Thanks!
=K х 7. The temperature T(2,t) in an insulated rod of length L and diffusivity k is given by the heat equation ОТ 22T 0 < x < L. at Əx2' Initially this rod is at constant temperature To, and immediately after t=0 the temperature at x = L is suddenly increased to T1. The temperature at x =...
Partial differential equation-wave equation
One end of the pipe is closed, which corresponds to the boundary conditionu(0, t) = 0, for t > 0. The other end of the pipe is open, which correspondsto the boundary condition ux(L, t) = 0, for t > 0.(a) Suppose that µ < 0, so µ = −k^2 for some k > 0. Find the non-trivial solution X(x) that satisfies equations (3), stating clearly what values k is allowed to take.(b) Write down the general solution of equation...
(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...
The conductive heat transfer in a rod of length L is described by the equation au...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
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)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive in...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis....
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis. Let u(x, t) be the temperature. Consider the following conditions. (A) There is heat transfer from the lateral surface of the rod into the surrounding medium, which is held at temperature 0° (B) There is heat transfer from the left end into the surrounding medium, which is held at a constant temperature of 0° (C) The left end is insulated. (D) The right end...
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1....
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1. of ordinary differential equations. (b) Suppose the above partial differential equation has boundary condition uz (0,t) 0, u(20, t)0. Use separations of variables to determine the corresponding bound- ary conditions that the ordinary differential equations found in (a) must satisfy. (c) (Yes or no) Could the partial differential equation, u -2uzt-5utt, be separated into two ordinary differential equations?
(a) Use separation of variables to...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
d1=7 d2=8 Any help would be greatly appreciated. Question 3 Left end (r-0) of a copper rod of length 100mm is kept at a constant temperature of Temp-1 0 a 2 degrees and the right end and sides are in...
d1=7
d2=8
Any help would be greatly appreciated.
Question 3 Left end (r-0) of a copper rod of length 100mm is kept at a constant temperature of Temp-1 0 a 2 degrees and the right end and sides are insulated, so that the temperature in the ul ul where D = 111 mm2/s for copper. rod, u(x,t), obeys the heat partial DE, Ot Ox (a) Write the boundary conditions for il(x,t) of the problem above. Note that for the left...
This is a question about
Partial differential equation - Heat equation. Please help solving
part (a) and show clear explanations. Thanks!
=K х 7. The temperature T(2,t) in an insulated rod of length L and diffusivity k is given by the heat equation ОТ 22T 0 < x < L. at Əx2' Initially this rod is at constant temperature To, and immediately after t=0 the temperature at x = L is suddenly increased to T1. The temperature at x =...
(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...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
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)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Problem #6: A rod of length I coincides with the interval [0, L] on the x-axis. Let u(x, t) be the temperature. Consider the following conditions. (A) There is heat transfer from the lateral surface of the rod into the surrounding medium, which is held at temperature 0° (B) There is heat transfer from the left end into the surrounding medium, which is held at a constant temperature of 0° (C) The left end is insulated. (D) The right end...
(a) Use separation of variables to rewrite the partial differential equation below into a pair 1. of ordinary differential equations. (b) Suppose the above partial differential equation has boundary condition uz (0,t) 0, u(20, t)0. Use separations of variables to determine the corresponding bound- ary conditions that the ordinary differential equations found in (a) must satisfy. (c) (Yes or no) Could the partial differential equation, u -2uzt-5utt, be separated into two ordinary differential equations?
(a) Use separation of variables to...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
d1=7
d2=8
Any help would be greatly appreciated.
Question 3 Left end (r-0) of a copper rod of length 100mm is kept at a constant temperature of Temp-1 0 a 2 degrees and the right end and sides are insulated, so that the temperature in the ul ul where D = 111 mm2/s for copper. rod, u(x,t), obeys the heat partial DE, Ot Ox (a) Write the boundary conditions for il(x,t) of the problem above. Note that for the left...
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