Numerical Technics In Engineering

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ow problem. Consider a thin steel plate to be a10 20 (cm)2 rectangle. If one side of the 10 cm edge is held at 1000C and the otherthree edges are held at 00C, what are the steady-state temperature at interior points?We can state the problem mathematically in this way if we assume that heat owsonly in the x and y directions:Find u(x; y) (temperature) such that@2u@x2 +@2u@y2 = 0 (3)with boundary conditionsu(x; 0) = 0u(x; 10) = 0u(0; y) = 0u(20; y) = 100We replace the di erential equation by a di erence equation1h2 [ui+1;j + ui????1;j + ui;j+1 + ui;j????1 ???? 4ui;j ] = 0 (4)5which relates the temperature at the point (xi; yj) to the temperature at four neigh-bouring points, each the distance h away from (xi; yj ). An approximation of Equation(3) results when we select a set of such points (these are often called as nodes) and nd the solution to the set of di erence equations that result.(a) If we choose h = 5 cm , nd the temperature at interior points.(b) Write a program to calculate the temperature distribution on interior points withh = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Discuss your solutions andexamine the e ect of grid size h.(c) Modi ed the di erence equation (4) so that it permits to solve the equation@2u@x2 +@2u@y2 = xy(x ???? 2)(y ???? 2)on the region0 x 2; 0 y 2with boundary condition u = 0 on all boundaries except for y = 0, where u = 1:0.Write and run the program with di erent grid sizes h and discuss your numericalresults.
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