numerical technics in engineering

Here is a regular steady-aver heat  ow example. Consider a attenuated steel compound to be a 10 20 (cm)2 rectangle. If one party of the 10 cm margin is held at 1000C and the other three margins are held at 00C, what are the steady-aver sphere at inland subject-matters? We can aver the example mathematically in this way if we wear 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 stipulation predicaments u(x; 0) = 0 u(x; 10) = 0 u(0; y) = 0 u(20; y) = 100 We re-establish 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 sphere at the subject-matter (xi; yj) to the sphere at impure neigh- bouring subject-matters, each the interval h far from (xi; yj ). An avenue of Equation (3) developments when we prime a set of such subject-matters (these are frequently determined as nodes) and nd the breach to the set of dierence equations that development. (a) If we select h = 5 cm , nd the sphere at inland subject-matters. (b) Write a program to estimate the sphere distribution on inland subject-matters after a while h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Sift-canvass your breachs and examine the eect of grid greatness h. (c) Modied the dierence equation (4) so that it permits to clear-up the equation @2u @x2 + @2u @y2 = xy(x ???? 2)(y ???? 2) on the region 0 x 2; 0 y 2 after a while stipulation predicament u = 0 on all boundaries save for y = 0, where u = 1:0. Write and run the program after a while dierent grid greatnesss h and sift-canvass your numerical results.