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Partial Differentials Notes_Part_15

Partial Differentials Notes_Part_15 - for the planar heat...

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for the planar heat equation is given by the linear superposition formula u ( t, x, y ) = 1 4 πγ t integraldisplayintegraldisplay f ( ξ, η ) e [( x ξ ) 2 +( y η ) 2 ] / (4 γ t ) dξ dη. (12 . 125) We can interpret the solution formula (12.125) as a two-dimensional convolution u ( t, x, y ) = F ( t, x, y ) 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 πγ t e [ x 2 + y 2 ] / (4 γ 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 , 0 , otherwise , then the solution can be evaluated using (12.125), as follows: u ( t, x, y ) = 1 4 π t integraldisplayintegraldisplay D e [( x ξ ) 2 +( y η ) 2 ] / (4 t ) dξ dη, where the integral is over the unit disk D = { ξ 2 + η 2 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. 12.6. The Planar Wave Equation. Let us next consider the two-dimensional wave equation 2 u ∂t 2 = c 2 Δ u = c 2 parenleftbigg 2 u ∂x 2 + 2 u ∂y 2 parenrightbigg , (12 . 128) which models the unforced vibrations of a homogeneous two-dimensional membrane, e.g., a
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