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**Unformatted text preview: **for the planar heat equation is given by the linear superposition formula u ( t, x, y ) = 1 4 Ï€Î³ t integraldisplayintegraldisplay f ( Î¾, Î· ) e âˆ’ [( x âˆ’ Î¾ ) 2 +( y âˆ’ Î· ) 2 ] / (4 Î³ t ) dÎ¾ dÎ·. (12 . 125) We can interpret the solution formula (12.125) as a two-dimensional convolution u ( t, x, y ) = F ( t, x, y ) âˆ— f ( x, y ) (12 . 126) of the initial data with a one-parameter family of progressively wider and shorter Gaussian filters F ( t, x, y ) = F ( t, x, y ; 0 , 0) = 1 4 Ï€Î³ t e âˆ’ [ x 2 + y 2 ] / (4 Î³ t ) . (12 . 127) As in (8.51), such a convolution can be interpreted as a weighted averaging of the function, and has the effect of smoothing out the initial signal f ( x, y ). Example 12.14. If our initial temperature distribution is constant on a circular region, say u (0 , x, y ) = braceleftbigg 1 x 2 + y 2 < 1 , , otherwise , then the solution can be evaluated using (12.125), as follows: u ( t, x, y ) = 1 4 Ï€ t integraldisplay integraldisplay D e âˆ’ [( x âˆ’ Î¾ ) 2 +( y âˆ’ Î· ) 2 ] / (4 t ) dÎ¾ dÎ·, where the integral is over the unit disk D = { Î¾ 2 + Î· 2 â‰¤ 1 } . Unfortunately, the integral cannot be expressed in terms of elementary functions. On the other hand, numerical evaluation of the integral is straightforward. A plot of the resulting radially symmetric solution appears in Figure 12.8. One could also interpret this solution as the diffusion of an animal population in a uniform, isotropic environment, or bacteria in a similarly uniform large petri dish, that is initially confined to a circular region....

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