Wiener–Hopf method: Difference between revisions

Revision as of 16:05, 16 March 2019

Revision as of 16:13, 16 March 2019

Line 20:

:<math>\boldsymbol{L}_{xy}f(x,y)=0,</math>

where <math>\boldsymbol{L}_{xy}</math> is a linear operator which contains derivatives with respect to {{mvar|x}} and {{mvar|y}}, subject to the mixed conditions on {{mvar|y}}&nbsp;=&nbsp;0, for some prescribed function {{math|''g''(''x'')}},

:<math>f=g(x)\text{ for }x\leq 0, \quad f_{y}f_y=0\text{ when }x>0</math>

and decay at infinity i.e. {{mvar|f}}&nbsp;→&nbsp;0 as <math>\boldsymbol{x}\rightarrow \infty</math>.

Taking a [[Fourier transform]] with respect to {{mvar|x}} results in the following [[ordinary differential equation]]

: <math>\boldsymbol{L}_y \widehat{f}(k,y)-P(k,y)\hatwidehat{f}(k,y)=0,</math>

where <math>\boldsymbol{L}_{y}</math> is a linear operator containing {{mvar|y}} derivatives only, {{math|''P''(''k,y'')}} is a known function of {{mvar|y}} and {{mvar|k}} and

: <math> \widehat{f}(k,y)=\int_{-\infty}^\infty f(x,y)e^{-ikx} \, \textrm{d}x. </math>

If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted {{math| ''F''(''k'',''y'')}}, a general solution can be written as

: <math> \hatwidehat{f}(k,y)=C(k)F(k,y), </math>

where {{math|''C''(''k'')}} is an unknown function to be determined by the boundary conditions on {{mvar|y}}=0.

Line 40 ⟶ 39:

: <math> \widehat{g}(k)+\widehat{f}_{+}(k,0) = \widehat{f}_{-}(k,0)+\widehat{f}_{+}(k,0) = \widehat{f}(k,0) = C(k)F(k,0) </math>

and, on taking derivatives with respect to <math>y</math>,

: <math> \hatwidehat{f}'_{-}(k,0) = \widehat{f}'_{-}(k,0)+\widehat{f}'_{+}(k,0) = \widehat{f}'(k,0) = C(k)F'(k,0). </math>

Eliminating <math>C(k)</math> yields