KKKQ 3013 PENGIRAAN BERANGKA

1. KKKQ 3013PENGIRAAN BERANGKA Assignment #6 – Partial Differential Equations 26 September 2007 8.00 pm – 10.00 pm

2. Assignment #6 (adapted courtesy of ref. [1]) Laplace equation A rectangular plate (with a circular corner) is insulated at its right edge. The plate is to be heated for a test run. In order to find the steady state temperature distribution on the plate, a centred finite divided difference have been suggested, with the plate divided into grids as shown above (where each grid Dx = Dy = 20mm). If x’ = 0.732 Dx and y’ = 0.732 Dy, determine all the unknown steady-state temperature on the plate (given that the steady state temperature on the uninsulated edge are as shown above). Employ a Gauss-Seidel method with a relaxation parameter l = 1.5. Iterate until an accuracy, where the approximate relative error |ea| < 1%, is achieved. [Hint: 1. For grids at the insulated edge, similar 2nd order centered finite difference for the laplace equation should also be obtained by using those grids as the center. In addition, the boundary condition dT/dx = 0 could be expressed in centered finite difference, to help substitute/reduce the unknown temperatures. 2. At point (1,1) derive the laplace equation based from the basic idea/principle for obtaining a numerical 2nd order difference] [1] Chapra, S.C & Canale, R.P, Numerical Methods for Engineers, McGraw-Hill 5th ed. (2006) Week 12