Heat transfer in a rectangle

The methodology is illustrated using a Laplace equation for heat transfer in a rectangle[7] [8] using length L and height H. The governing equation for the temperature in dimensionless form can be written as [8] 

For convenience, let (, = —. This converts the governing equation (equation 6.6) 

In equation (6.11), there are N second order equations. These are converted to 2N first order equations as follows (Subramanian White, 2000b [9] Rice and Do, 1995 [10] see chapter 2.1.2)  

In equation (6.14), the unknown constants ci, i = 1..N are found after integrating the equations in (6.12) and by using the boundary conditions at y = 1. 

6 Method of Lines for Elliptic Partial Differential Equations 

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Example 6.6. Numerical Solution for Heat Transfer in a Rectangle... 

Example 6.1 (heat transfer in a rectangle) is solved again using the numerical method of lines. The procedure involved in solving a linear or nonlinear steady state elliptic PDE numerically is summarized as follows ... 

Example 7.4, heat transfer in a rectangle with a sinusoidal initial profile,[2] is solved here again using the Laplace transform technique. The dimensionless temperature profile is governed by ... 

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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