Application to problems on the conduction of heat

The reader knows that ordinary and partial differential. equations, differ in this respect Whilft ordinary differential cgimtiong ha v only a finite number of independent particular integrals, partial differential equations have an infinite number- of such integrals. And in practical work we have to pick out one particular integral to satisfy the conditions under which any given experiment is performed. Suppose that a value of V is required in the equation 

Multiply by 1/tt and find the limits when a has different values between 0 and oo. Hence, from (30) and (32), page 479, we have the particular solution satisfying the required conditions 

Examples.—(1) A large iron plate rr cm. thick and at a uniform temperature of 100° is suddenly placed in a bath at zero temperature for 10 seconds. Required the temperature of the middle of the plate at the end of 10 seconds, supposing that the difiusivity k of the plate is 0 2 C.O.S. units, and that the 

We must substitute the coefficients of this series for av a2, o3,. .. in (7), to get a solution satisfying dll the required conditions. Note 03, a4. in (9) are zero. We thus obtain the required solution 

To introduce the numerical data. When x = fyv, t = 10, c = 0-2. Hence use a table of logarithms. The result is accurate to the tenth of a degree if all terms of the series other than the first be suppressed. Hence use 

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