Partial Differential Equations- Two-Dimensional Heat Equation

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using separation of variables to ?nd the temperature distribution of rectangular and circular plates in the transient state.
Suppose that for a thin rectangular plate which occupies the plane region 0 ? x ? a, 0 ? y ? b, the top and bottom faces are insulated, and that its four edges are held at zero temperature. If the plate has the initial temperature function u(x, y, 0) = f(x, y), then in the transient state its temperature function u(x, y, t) is the solution of the following initial boundary value problem

We shall ?nd the solution of (36.1)