Transient Heat Transfer in a Rectangle

Consider heat transfer in a rectangle with a derivative boundary condition. The dimensionless temperature profile is governed by  

8 Laplace Transform Technique for Partial Differential Equations 

In examples 8.3 and 8.4 Maple was used to invert from the Laplace domain to the time domain. Unfortunately, these two examples are very simple and, hence, we could invert to the time domain using Maple. For practical problems, inversion is not straightforward. The inversion to the time domain can be done in two different ways. In section 8.1.4, short time solutions will be obtained by converting the solution in Laplace domain to an infinite series. In section 8.1.5, a long time solution will be obtained by using the Heaviside expansion theorem. 

4 Laplace Transform Technique for Parabolic Partial Differential Equations - Short Time Solution 

The methodology is the same as that used in section 8.1.3. When Maple fails to invert the Laplace domain solution to the time domain, a short time solution can be obtained by converting the Laplace domain solution to an infinite series in which each term can be easily inverted to time domain. The solution obtained for heat transfer in a rectangle in example 7.1 using the separation of variables method cannot be used at short times. At time t = 0, one would need infinite number of terms in the separation of variables solution. Fortunately, the Laplace transform technique helps us obtain a solution, which can be used efficiently at short times also. This is best illustrated with the following examples. 

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