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notes 10-5

# notes 10-5 - problem we found that a solution of this heat...

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SECTION 10.5 SEPARATION OF VARIABLES AND THE HEAT EQUATION, AGAIN EXAMPLE. Determine whether the method of separation of variables can be used to replace the following partial differential equation by a pair of ordinary differential equations. If so, find the equations. u xx + u xt + u t = 0

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Suppose a uniform rod of length L has insulated sides and ends which are maintained at 0 C. Suppose also that at time t = 0 the temperature at point x along the rod is given by a function f ( x ) and finally that the temperature at point x , time t is given by u ( x, t ). Using the physical properties of temperature and heat transfer one can show (demonstration begins on page 649) that u ( x, t ) must satisfy α 2 u xx = u t , u ( x, 0) = f ( x ) , and u (0 , t ) = u ( L, t ) = 0 for t 0 , where the constant α 2 is called the thermal diffusivity and depends on the makeup of the rod. Using centimeters and seconds, α 2 ranges from 0.00144 for water through 0.011 for granite to 1.71 for silver. There’s a table on page 604. By separating variables and then solving a differential equation and a boundary value
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Unformatted text preview: problem, we found that a solution of this heat conduction problem is given by u ( x,t ) = ∞ X n =1 c n e-n 2 π 2 α 2 t L 2 sin nπx L , where the coeﬃcients c n are chosen so that ∞ X n =1 c n sin nπx L = f ( x ) = u ( x, 0) for 0 ≤ x ≤ L. We now know how to ﬁnd the sine series for f ( x ) on 0 ≤ x ≤ L so we are in business! EXAMPLE. Find the solution of the heat conduction problem . 01 u xx = u t , < x < 100 , t > 0; u (0 ,t ) = u (100 ,t ) = 0 , t > 0; u ( x, 0) = 3 sin πx-5 sin 4 πx + 2 sin 17 πx 2 , ≤ x ≤ 100 . EXAMPLE. Consider the conduction of heat in a rod 80 cm long whose sides are insulated and whose ends are maintained at 0 ◦ C for all t > 0. Find an expression for the temperature u ( x,t ) if the initial temperature distribution in the rod is given below. Suppose that α 2 = 1. u ( x, 0) = ≤ x < 20 30 20 ≤ x ≤ 60 60 < x ≤ 80 HOMEWORK: SECTION 10.5...
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