Graetz Problem-Finite Difference Solution

Example 3.2.14. Graetz Problem-Finite Difference Solution 

The Graetz problem (heat or mass transfer) in cylindrical coordinates with parabolic velocity profile is solved here. The governing equation for the eigenfunction is [15] [8] 

Equation (3.82) is a second order equation with two boundary conditions (equations (3.83) and (3.84). In equation (3.82), A, is the eigenvalue. To solve for 

equation (3.82) is discretized using finite differences as in section 3.2.3. This yields N equations for the interior node points. The boundary conditions (equations (3.83) and (3.84) are converted to finite difference form. This yields two equations. There are a total of N+2 node points including the boundaries. There are N+2 dependent variables (yi, = o.,n+i)- There is an additional variable X. The additional equation is (3.85). Hence, there are N+3 variables (yi,i = o..n+i and A-) to be solved from N+3 equations. There are infinite solutions for the differential equation (3.82). Hence, there are multiple solutions for the system of finite difference equations. This example is solved in Maple below  

The central difference expressions for the second and first derivatives are 

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