# numerical technics in engineering

Here is a normal steady-aver heat  ow tenor. Consider a slim steel platter to be a 10 20 (cm)2 rectangle. If one margin of the 10 cm party is held at 1000C and the other three partys are held at 00C, what are the steady-aver region at inland sharp-ends? We can aver the tenor mathematically in this way if we presume that heat  ows only in the x and y directions: Find u(x; y) (temperature) such that @2u @x2 + @2u @y2 = 0 (3) delay designation moods 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 region at the sharp-end (xi; yj) to the region at impure neigh- bouring sharp-ends, each the length h detached from (xi; yj ). An similarity of Equation (3) terminations when we fine a set of such sharp-ends (these are repeatedly named as nodes) and nd the discontinuance to the set of dierence equations that termination. (a) If we select h = 5 cm , nd the region at inland sharp-ends. (b) Write a program to weigh the region distribution on inland sharp-ends delay h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Argue your discontinuances and examine the eect of grid dimension 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 delay designation mood u = 0 on all boundaries save for y = 0, where u = 1:0. Write and run the program delay dierent grid dimensions h and argue your numerical results.