4 fourier transforms of fundamental solutions we have

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4. Fourier Transforms of Fundamental Solutions We have so far dealt only with PDEs defined over the positive half line. When the domain of the PDE is the whole of R , we can re- cover fundamental solutions by inverting a Fourier transform. We il- lustrate the procedure for an interesting subclass of the equations we have looked at. Consider as motivation the heat equation. This has a well known symmetry ˜ u ² ( x, t ) = 1 1 + 4 ²t exp - ²x 2 1 + 4 ²t u x 1 + 4 ²t , t 1 + 4 ²t . (4.1) Take u = 1 . The general method we have introduced implies that we should look for a fundamental solution K ( x, y, t ) with the property that Z -∞ e - ²y 2 K ( x, y, t ) dy = 1 1 + 4 ²t exp - ²x 2 1 + 4 ²t . (4.2) It is easy to verify that K ( x, y, t ) = 1 4 πt e - ( x - y ) 2 4 t is a solution of this equation. However, if we did not know K to begin with, it is not clear how it can be extracted from (4.2). This integral equation does not even have a unique solution, since for ² > 0 and any bounded,
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FUNDAMENTAL SOLUTIONS 21 continuous odd function h we have R -∞ e - ²y 2 ( K ( x, y, t ) + h ( y )) dy = R -∞ e - ²y 2 K ( x, y, t ) dy. A method for extracting the heat kernel from (4.1) using two station- ary solutions is given in [7]. However, the heat equation also possesses a symmetry of the form ˜ u ² ( x, t ) = e - ²x + ² 2 t u ( x - 2 ²t, t ) . If we take ² = and let u = 1, then this symmetry leads to Z -∞ e - iλy K ( x, y, t ) = e - iλx - λ 2 t . (4.3) So we have obtained the Fourier transform of the heat kernel from a Lie group symmetry. The Fourier inversion theorem easily gives the one dimensional heat kernel. This method applies to more problems than the heat equation. When- ever we have a PDE u t = σu xx + f ( x ) u x - g ( x ) u, x R (4.4) with a six dimensional Lie algebra of symmetries, we may construct Fourier transforms of fundamental solutions. We have seen conditions on the drift f which guarantee the existence of a nontrivial symmetry group. Proposition 4.1 is a special case of our earlier results, which we have not yet exploited. It guarantees the existence of a six dimen- sional symmetry group for equations of the form (4.4) and provides the symmetries that will be used to produce Fourier transforms. Proposition 4.1. Let the drift function f in equation (4.4) satisfy the Riccati equation σf 0 + 1 2 f 2 + 2 σg = 1 2 Ax 2 + Bx + C (4.5) where A, B, C are arbitrary constants. Then equation (4.4) has a six dimensional Lie algebra of symmetries. Moreover if A 6 = 0 , then it has a symmetry of the form ˜ u ² ( x, t ) = e - σ cosh( At ) x + 2 2 σ sinh(2 At )+ σ A (1 - cosh( At )) × e 1 2 σ ( F ( x - 2 ² sinh( At )) - F ( x ) ) u ( x - 2 ² sinh( At ) , t ) . (4.6) If f satisfies the special case σf 0 + 1 2 f 2 + 2 σg = Ax + B (4.7) then it has a symmetry of the form ˜ u ² ( x, t ) = e - ²x 2 σ + ² 2 t 4 σ - 4 σ t 2 + 1 2 σ ( F ( x - ²t ) - F ( x )) u ( x - ²t, t ) . (4.8) In both cases F 0 ( x ) = f ( x ) .
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22 MARK CRADDOCK Proof. Let v = ξ∂ x + τ∂ t + φ∂ u . Lie’s method shows that v generates symmetries if and only if ξ = 1 2 t + ρ , φ ( x, t, u ) = α ( x, t ) u where α = - x 2 8 σ τ tt - x 2 σ ρ t - 1 4 σ ( xf ( x )) τ t - 1 2 σ ρ + η (4.9) and - x 2 8 σ τ ttt - x 2 σ ρ tt + η t = - 1 4 τ tt - 1 2 σ ( σ ( xf ) 00 + f ( xf ) 2 + 2( σxg ) 0 ) τ t - 1 2 σ ( σf 00 + ff 0 + 2 σg 0 ) ρ. (4.10) The Lie algebra of symmetries is six dimensional if and only if ρ is nonzero. This occurs when f satisfies the given Riccati equations. In
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