PDE. Please show all steps in detail.
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2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endp...
PDE. Please show all steps in detail. 2. Consider the 1D heat equation in a rod...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<<t (a) Set up the initial-boundary value problem modeling this scenario. (b) Set up and...
Use the 1D diffusion equation to find T(x,t): 2. A bar of length L has an...
Use the 1D diffusion equation to find T(x,t):
2. A bar of length L has an initial temperature distribution T(x,0) = a + br ( x = L are the ends of the bar). The ends are insulated. Find T(x,t) for t > 0. 0 and си oFlu сх
For (1) – (3), the model is with regards to a rod of length L with...
For (1) – (3), the model is with regards to a rod of length L with thermal diffusivity k coinciding along the interval (0, L) on the z-axis. Set up the boundary-value problem for the temperature u(x,t). (1) The left end is insulated and the right end is held at a temperature of 0°. The initial temperature is 1° throughout. (2) The left end is at a temperature of 50e-t, the right end if held at zero, and there is...
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...
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 maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
Partial Differential Equations Question: A homogeneous cylindrical rod of length L = 1 is insulated along...
Partial Differential Equations Question: A homogeneous cylindrical rod of length L = 1 is insulated along the cylindrical side. At the end caps, heat exchange obeys Newton’s law of cooling, i.e. the flux is proportional to the difference of the temperature of the rod with that of the surrounding medium, written explicitly as ux(0,t) = u(0,t ) -T1 and ux(1,t) = T2- u(1,t) where T1 = 0 and T2 = 1. Find the steady state distribution of the temperature.
Suppose heat is lost from the lateral surface of a thin rod of length L into...
Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form du - hu- az ar 0<x<L, t > 0, ha constant. Find the temperature uix, t) if the initial temperature is fx) throughout and the ends 0 and XL are insulated. See the figure u(x, t) *)-(wax) ). 2 [(? I'moscoap 90.cr)()+(-*...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends insulated for all time (zero Neumann boundary conditions), if the initial temperature is given by (2) = x+sin ax. First, formulate the mathematical problem and complete the three steps as described. Mathematical Formulation Step 1: Derive an expression for all nontrivial product (separated) solutions including an eigenvalue problem satisfying the boundary conditions Step 2: Solve the eigenvalue problem Step 3: Use the superposition principle...
All the computer work is done. Need help with hand written solution for all questions, thanks....
All the computer work is done. Need help with hand written
solution for all questions, thanks.
Freezing pipes. The Problem and Model. The project is to determine an approximation for the indoor temperature u(t) in an unheated builg. The model uses Newton's cooling law, insulation data k and a formula for the ambient outside temperature A(t) (see the background section infra) u'(t) + ku(t) = kA(t) u(0)o Assumptions and Notation. Let the daily temperature in Salt Lake City vary from...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal properties (assume k=1): - xe" (0 < x <l, t>0) 04 (0,1) = 0 (1,1)= 1 a) Determine the equilibrium temperature distribution, and plot it on the interval [0, 1] b) Where does the energy enter, and where does it leave the rod? Explain your answers.
Suppose that a death body is found inside a room that is kept at a constant 65 F. The body's temp...
Suppose that a death body is found inside a room that is kept at a constant 65 F. The body's temperature was first measured to be 800 F at 7:00 am, and an hour later 785' F. Assume that the person was healthy before she/he died and had a body temperature of 98.6 F.Use the Newton's Law of Cooling Model to determine the time when the person died. (a) Write the corresponding initial value problem describing the situation (b) Find...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<<t (a) Set up the initial-boundary value problem modeling this scenario. (b) Set up and...
Use the 1D diffusion equation to find T(x,t):
2. A bar of length L has an initial temperature distribution T(x,0) = a + br ( x = L are the ends of the bar). The ends are insulated. Find T(x,t) for t > 0. 0 and си oFlu сх
For (1) – (3), the model is with regards to a rod of length L with thermal diffusivity k coinciding along the interval (0, L) on the z-axis. Set up the boundary-value problem for the temperature u(x,t). (1) The left end is insulated and the right end is held at a temperature of 0°. The initial temperature is 1° throughout. (2) The left end is at a temperature of 50e-t, the right end if held at zero, and there is...
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 maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
Suppose heat is lost from the lateral surface of a thin rod of length L into a surrounding medium at temperature zero. If the linear law of heat transfer applies, then the heat equation takes on the form du - hu- az ar 0<x<L, t > 0, ha constant. Find the temperature uix, t) if the initial temperature is fx) throughout and the ends 0 and XL are insulated. See the figure u(x, t) *)-(wax) ). 2 [(? I'moscoap 90.cr)()+(-*...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends insulated for all time (zero Neumann boundary conditions), if the initial temperature is given by (2) = x+sin ax. First, formulate the mathematical problem and complete the three steps as described. Mathematical Formulation Step 1: Derive an expression for all nontrivial product (separated) solutions including an eigenvalue problem satisfying the boundary conditions Step 2: Solve the eigenvalue problem Step 3: Use the superposition principle...
All the computer work is done. Need help with hand written
solution for all questions, thanks.
Freezing pipes. The Problem and Model. The project is to determine an approximation for the indoor temperature u(t) in an unheated builg. The model uses Newton's cooling law, insulation data k and a formula for the ambient outside temperature A(t) (see the background section infra) u'(t) + ku(t) = kA(t) u(0)o Assumptions and Notation. Let the daily temperature in Salt Lake City vary from...
2. (16pts) Consider the following heat equation for a rod of length L=1 with constant thermal properties (assume k=1): - xe" (0 < x <l, t>0) 04 (0,1) = 0 (1,1)= 1 a) Determine the equilibrium temperature distribution, and plot it on the interval [0, 1] b) Where does the energy enter, and where does it leave the rod? Explain your answers.
Suppose that a death body is found inside a room that is kept at a constant 65 F. The body's temperature was first measured to be 800 F at 7:00 am, and an hour later 785' F. Assume that the person was healthy before she/he died and had a body temperature of 98.6 F.Use the Newton's Law of Cooling Model to determine the time when the person died. (a) Write the corresponding initial value problem describing the situation (b) Find...
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