numerical technics in engineering

Here is a natural steady-specify heat  ow drift. Consider a attenuated steel plate to be a 10 20 (cm)2 rectangle. If one laterality of the 10 cm cause is held at 1000C and the other three causes are held at 00C, what are the steady-specify weather at internal tops? We can specify the drift mathematically in this way if we take that heat  ows only in the x and y directions: Find u(x; y) (temperature) such that @2u @x2 + @2u @y2 = 0 (3) after a while designation terms u(x; 0) = 0 u(x; 10) = 0 u(0; y) = 0 u(20; y) = 100 We restore the dierential equation by a dierence equation 1 h2 [ui+1;j + ui????1;j + ui;j+1 + ui;j????1 ???? 4ui;j ] = 0 (4) 5 which relates the weather at the top (xi; yj) to the weather at disgusting neigh- bouring tops, each the separation h detached from (xi; yj ). An way of Equation (3) upshots when we choice a set of such tops (these are frequently named as nodes) and nd the reresolution to the set of dierence equations that upshot. (a) If we prefer h = 5 cm , nd the weather at internal tops. (b) Write a program to proportion the weather classification on internal tops after a while h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Sift-canvass your resolutions and examine the eect of grid bigness h. (c) Modied the dierence equation (4) so that it permits to explain the equation @2u @x2 + @2u @y2 = xy(x ???? 2)(y ???? 2) on the region 0 x 2; 0 y 2 after a while designation term u = 0 on all boundaries save for y = 0, where u = 1:0. Write and run the program after a while dierent grid bignesss h and sift-canvass your numerical results.