hw4_soln_rough

# hw4_soln_rough - w x t gives w t x t-kw xx x t = u t t x V...

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Problems from 2.4 of Strauss: 16. Set v ( x, t ) = e bt u ( x, t ). By computation, if u ( x, t ) solves the heat equation with constant dissipation and initial data u ( x, 0) = φ ( x ), then v ( x, t ) solves the standard heat equation v t ( x, t ) = kv xx ( x, t ) v ( x, 0) = φ ( x ) By formula, we know that v ( x, t ) = 1 4 πkt Z -∞ e - ( x - y ) 2 / 4 kt φ ( y ) dy Hence u ( x, t ) = e - bt v ( x, t ) = e - bt 4 πkt Z -∞ e - ( x - y ) 2 / 4 kt φ ( y ) dy. 18. Set w ( x, t ) = u ( x + tV, t ). By computation, if u ( x, t ) solves the heat equa- tion with convection, applying the heat operator to
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Unformatted text preview: w ( x, t ) gives w t ( x, t )-kw xx ( x, t ) = u t ( t, x ) + V u x ( x + tV, t )-ku xx ( x, t ) = 0 w ( x, 0) = u ( x, 0) = φ ( x ) . Hence w ( x, t ) = 1 √ 4 πkt Z ∞-∞ e-( x-y ) 2 / 4 kt φ ( y ) dy. However, by deﬁnition of w , u ( x, t ) = w ( x-V t, t ) and hence u ( x, t ) = w ( x-V t, t ) = 1 √ 4 πkt Z ∞-∞ e-( x-V t-y ) 2 / 4 kt φ ( y ) dy. 1...
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