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11. Show that if u(x, t) = (f ∗ Ht)(x) where Ht is th...

11. Show that if u(x, t) = (f ∗ Ht)(x) where Ht is the heat kernel, and f is Riemann integrable, then Z 1 0 |u(x, t) − f(x)| 2 dx → 0 as t → 0. 2. Let f and g be the functions defined by f(x) = χ[−1,1](x) = ½ 1 if |x| ≤ 1, 0 otherwise, and g(x) = ...

Additional problem 1. Solve the following initial-boundary v...

Additional problem 1. Solve the following initial-boundary value problem for the heat equation ut = kuxx, x ∈ [0, L], t ∈ [0, ∞) u(0, t) = A, u(L, t) = B for t ∈ [0, ∞] u(x, 0) = f(x), for x ∈ [0, L], where A and B are constants, in general nonzero. Since we know...

Heat! Temperature: Many people consider themselves to be e...

Heat! Temperature: Many people consider themselves to be excellent judges of temperature. “It’s cold outside” is something I’m sure we have all heard, but in physics we like to be fairly precise. A fair question to the person making the statement above might be, “wh...

Heat Transfer In this project, you will perform analysis on...

Heat Transfer In this project, you will perform analysis on a two-dimensional fin. The fin is made of aluminum (k = 200 W/mK, ρ = 2700 kg/m3 , cp = 890 J/kgK, α = 8.4x10-5 m2 /s and is 5 cm long and 1 cm thick. The fin is initially at a uniform temperature of 250 deg C, when it is exp...

1) The temperature of a bar of an unknown metal rises by 10....

1) The temperature of a bar of an unknown metal rises by 10.0°C when it absorbs 0.527 kJ of energy by heat. The mass of the bar is 525 g. Determine the specific heat of the unknown. 2) Lake Erie contains roughly 4.00 ✕ 1011 m3 of water. (a) How much energy is required to raise the temperatu...

Review (critically) the Heat sink article on Wikipedia. T...

Review (critically) the Heat sink article on Wikipedia. This is a nice article that has both general information and some specific heat transfer analysis of how and why heat sinks work. Wikipedia has improved greatly in the last 15 years in solidifying its technical content (that’s the reas...

Heat Transfer Determine the steady temperature inside a 90 ...

Heat Transfer Determine the steady temperature inside a 90 degree sector of a circular annulus ( 0<r<a , 0<theta <Pi/2), if its circular edge ( r =a) and one straight edge ( Theta = 0 ) are at 0 degrees throughout. The remaining straight edge ( Theta = Pi/2) is given as ...

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