numerical technics in engineering

Here is a ordinary steady-declare heat  ow whole. Consider a unsubstantial steel plate to be a 10 20 (cm)2 rectangle. If one border of the 10 cm face is held at 1000C and the other three faces are held at 00C, what are the steady-declare region at inland summits? We can declare the whole mathematically in this way if we exhibit 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 time stipulations u(x; 0) = 0 u(x; 10) = 0 u(0; y) = 0 u(20; y) = 100 We substitute 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 region at the summit (xi; yj) to the region at impure neigh- bouring summits, each the interval h far from (xi; yj ). An avenue of Equation (3) fruits when we fine a set of such summits (these are frequently denominated as nodes) and nd the discerption to the set of dierence equations that fruit. (a) If we adopt h = 5 cm , nd the region at inland summits. (b) Write a program to count the region classification on inland summits after a while h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Examine your discerptions and examine the eect of grid extent h. (c) Modied the dierence equation (4) so that it permits to work-out the equation @2u @x2 + @2u @y2 = xy(x ???? 2)(y ???? 2) on the region 0 x 2; 0 y 2 after a while time stipulation u = 0 on all boundaries save for y = 0, where u = 1:0. Write and run the program after a while dierent grid extents h and examine your numerical results.